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 grabbling 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. It's that 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.
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.