Yes, a mount can attempt to escape a grapple.

If rules are unclear, the DM must rule

Tarod's answer and nonymous' answer both argue oppositely as to whether the mounted rules or the grappling rules are the greater exception.

Both are missing a more important rule, under "Being the Dungeon Master":

Adjudicate the Rules. You oversee how the group uses the game’s rules, making sure the rules serve the group’s fun. You’ll want to read the rest of this chapter to understand those rules, and you’ll find the rules glossary essential.

But are the rules unclear?

Not to me. To me, obviously a grapple is the exception to the rule of mounted combat, so of course a mount can escape.

However, I can understand how someone might very reasonably argue the opposite.

But the answer doesn't hinge on which is the general and which is the exception.

It hinges on which makes sense.

And it's not "that RAW the rules say a mount cannot escape, but a DM is free to rule otherwise". RAW just doesn't say. The mounted combat rules provide simple rules for mounted combat and the grappling rules provide simple rules for grappling. Neither is going to account for every situation.

To consider a slightly different situation, a common trope in fiction involving horses is where a rider on a mount has the mount pull something out of somewhere. For instance, the cleric is stuck in the quicksand, the paladin throws them a rope, then uses the horse haul the cleric out.

To me, the answer isn't, "sorry, you can't do that, because your mount has "only three action options during that turn: Dash, Disengage, and Dodge". The answer is that the DM figures out how to model it.

The rules say:

Player characters and monsters can also do things not covered by these actions. Many class features and other abilities provide additional action options, and you can improvise other actions. When you describe an action not detailed elsewhere in the rules, the Dungeon Master tells you whether that action is possible and what kind of D20 Test you need to make, if any.

The rules serve you, you don't serve the rules

Does it make sense that a mount can't attempt to escape? No, it doesn't. Let's say a horse with a rider is stuck in a mud trap. Let's say the rules for this trap says a creature that failed its save is stuck in the mud and is grappled.

You don't need then to parse which rule is the exception to which rule.

Of course the horse can attempt to escape. How does the DM adjudicate the horse escaping the grapple? The grappling rules are right there.

You can't play D&D without a DM

The rules don't cover every situation. They are never going to.

It's not that the rules cover this situation, and the DM is free to rule otherwise.

D&D is a narrative story, the players use the rules to have a common understanding of how to resolve issues. The DM is there to apply the rules.

I don't even think this question comes up except in a hypothetical reading of the rules without a DM.

If you are a DM

If you're really actually unclear as to whether the horse can escape the grapple, make a decision. To me, it is so obvious the horse can escape the grapple that I would never think twice about it.

But if you really feel like it's more fun and makes more sense that the horse cannot escape, it's your game. Heck, there are probably circumstances where the rider is too heavy or something.

But don't you be trapped by a misreading of the rules. You can escape. So can the horse.

Not if the mount is Controlled (trained).

There are two types of mounts; domesticated or trained animals (Controlled Mounts) that are used to human interaction where the human is 'in charge,' and Independent mounts, which could be an impromptu Rhino in the wild that you jumped on, or it might just be a creature too intelligent that has its own ego, who won't suppress its natural survival instincts for the sake of behaving like a trained, subservient animal (but might still let you on for a ride).

Trained Mounts are a codified game term, but we don't exactly have a detailed, encyclopedic definition of all the things this entails, it simply says they are trained to accept a rider and move as you direct (which grappling stops). A warhorse, for example, will usually not buck and try to kick a rider during a massive melee combat with noise and bloodshed everywhere (exceptions exist and Handle Animal checks exist for them), or take a sudden galloping motion, unless and until the rider directs it to, because that's part of its training.

Likewise, you typically train your mounts to trot, canter, gallop, slow, stop, maybe even prance or jump, but not to "outwrestle that varmint."

Trained mounts generally remain passive except at the behest of their rider giving them a short list of commands that correlate to specific tasks. As long as the rider is controlling it, its training applies, and it only performs those tasks; it would suck if you wanted to charge forward and impale someone on a lance but the horse just decided to put its head down and eat. (It reasonably could, but because it's trained, it won't.)

Since RAW limits the actions you can take on a controlled mount to three things, the logic follows that it makes sense to assume that being a controlled mount means the creature is suppressing its own inclinations to take (a limited, trained for amount of) actions explicitly at the rider's discretion. This is why controlled mounts don't, for example, get attack someone in melee range that just stabbed them in an attempt to unmount the rider.

If, however, you choose to "let the reins free" (uncontrolled mount), the horse is no longer setting aside its natural reactions to obey its rider, and it can do anything it wants, from rearing back to strike with a hoof to jumping clear off the ground and spinning 360 while kicking its back legs.

Strictly RAW, no

As you said, a controlled mount has only three action options on its turn (see Controlling a Mount, PHB, p. 26).

Escaping a grapple specifically requires the Grappled creature to use its action to escape (PHB, p. 367):

Grappling

[...]

Ending a Grapple. A Grappled creature can use its action to make a Strength (Athletics) or Dexterity (Acrobatics) check against the grapple’s escape DC, ending the condition on itself on a success. The condition also ends if the grappler has the Incapacitated condition or if the distance between the Grappled target and the grappler exceeds the grapple’s range. In addition, the grappler can release the target at any time (no action required).

Because Dash, Disengage, and Dodge are the only actions a controlled mount can take, it cannot use its action to escape the grapple while it is controlled.

This is the strict reading of the rules, but your DM might rule it differently.

Side note

If the rider doesn't control the mount, it acts independently and is no longer limited to specific actions. In that case, it could use its action to attempt to escape a grapple.

Addendum: Why this is not a Specific vs General conflict

The Exceptions Supersede General Rules principle applies when two rules attempt to answer the same question in different ways. Here, they address different things.

• Grapple rule (general): A grappled creature can use its action to escape. This rule permits an action; it does not guarantee that the creature always has access to that action in every circumstance (i.e. a creature that is Incapacitated or Stunned can't escape a grapple even though the grapple rules say it can use its action to do so)

• Controlled mount rule (specific to controlled mounts): A controlled mount can take only Dash, Disengage, or Dodge. This rule restricts what actions are available in a very specific state: being a controlled mount.

There is no conflict between these rules. As a result, this is not a specific-vs-general issue. The grapple rules define the cost of escaping, while the controlled mount rules limit which actions are available. Since Escape is not an available action for a controlled mount, there is nothing for specific-over-general to resolve.

Yes

Exceptions Supersede General Rules.

Exceptions Supersede General Rules

General rules govern each part of the game. A general rule is in effect as long as something in the game doesn’t explicitly say otherwise. When an exception and a general rule disagree, the exception wins.

The Grappled condition is an exception to the rules for Mounted Combat because it applies in a narrower, triggered circumstance and explicitly grants an action, which mounted combat does not remove.

The Mounted Combat rules describe a general situation—how movement, initiative, and positioning work when one creature rides another. They do not restrict actions except where explicitly stated, and those restrictions apply to the mount’s normal actions.

By contrast, the Grappled condition is a defined mechanical state with explicit rules that apply only while the condition exists. Conditions are designed to override normal circumstances.

The Grappled condition explicitly grants a mechanical permission:

A grappled creature can use its action to make a Strength (Athletics) or Dexterity (Acrobatics) check against the grapple’s escape DC, ending the condition on itself on a success.

Because this permission is explicit and no mounted combat rule negates it, a grappled controlled mount may use its action to attempt to escape the grapple.

Only if the rider chooses to allow it to act independently.

While you’re mounted, you have two options. You can either control the mount or allow it to act independently. Intelligent creatures, such as dragons, act independently.

Bearing a rider puts no restrictions on the actions the mount can take, and it moves and acts as it wishes.

emphasis mine

Of course, if you allow it to act independently, your DM may choose to have the mount do something else entirely, though it's certainly plausible for an animal to resist the control of someone besides its rider. Arguably, a grappled mount may no longer be "controlled" in the sense that movement is now 0, precluding controlled actions. Ask your DM.