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RAW, no

As you said, a controlled mount has only three action options on its turn (see Controlling a Mount, PHB, p. 198).

Escaping a Grapple specifically requires the grappled creature to use its action to escape (PHB, p. 195):

A grappled creature can use its action to escape. To do so, it must succeed on a Strength (Athletics) or Dexterity (Acrobatics) check contested by your Strength (Athletics) check.

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: How to understand General vs Specific

The specific vs general 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.

Strictly RAW, no

As you said, a controlled mount has only three action options on its turn (see Controlling a Mount, PHB, p. 198).

Escaping a Grapple specifically requires the grappled creature to use its action to escape (PHB, p. 195):

A grappled creature can use its action to escape. To do so, it must succeed on a Strength (Athletics) or Dexterity (Acrobatics) check contested by your Strength (Athletics) check.

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 specific vs general 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.

RAW, no

As you said, a controlled mount has only three action options on its turn (see Controlling a Mount, PHB, p. 198).

Escaping a Grapple specifically requires the grappled creature to use its action to escape (PHB, p. 195):

A grappled creature can use its action to escape. To do so, it must succeed on a Strength (Athletics) or Dexterity (Acrobatics) check contested by your Strength (Athletics) check.

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.

General vs specific

The specific vs general 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.

RAW, no

As you said, a controlled mount has only three action options on its turn (see Controlling a Mount, PHB, p. 198).

Escaping a Grapple specifically requires the grappled creature to use its action to escape (PHB, p. 195):

A grappled creature can use its action to escape. To do so, it must succeed on a Strength (Athletics) or Dexterity (Acrobatics) check contested by your Strength (Athletics) check.

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: How to understand General vs Specific

The specific vs general 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.

RAW, no

As you said, a controlled mount has only three action options on its turn (see Controlling a Mount, PHB, p. 198).

Escaping a Grapple specifically requires the grappled creature to use its action to escape (PHB, p. 195):

A grappled creature can use its action to escape. To do so, it must succeed on a Strength (Athletics) or Dexterity (Acrobatics) check contested by your Strength (Athletics) check.

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.

RAW, no

As you said, a controlled mount has only three action options on its turn (see Controlling a Mount, PHB, p. 198).

Escaping a Grapple specifically requires the grappled creature to use its action to escape (PHB, p. 195):

A grappled creature can use its action to escape. To do so, it must succeed on a Strength (Athletics) or Dexterity (Acrobatics) check contested by your Strength (Athletics) check.

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.

General vs specific

The specific vs general 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.