Is 'a parents only appointment' correct?

It should be "a parents-only appointment", if you mean an appointment that is for parents only. When you have two Words acting as an adjective before a noun, you need a hyphen between the Words so that the meaning is clear.

For example, "two-Word modifiers" means modifiers that consist of two Words. However, "two Word modifiers" could mean two ordinary adjectives (like red and blue), since each could be considered a "Word modifier".