Pegasean

Yes, there is! That word is the adjective Pegasean. The OED gives it the following sense:

Of, relating to, or characteristic of Pegasus. Also: resembling Pegasus, esp. in his capacity for winged flight; (hence) poetically inspired or elevated.

It provides citations ranging between 1590 and 1988.

It notes that we most often now place the stress on the second syllable from the end, but that formerly we placed in on the third syllable from the end instead as attested by metrical evidence.

Not Pegasarian, at least not lately

People also tried to use Pegasarian, but this didn’t catch on and is now marked obsolete:

1607 E. Topsell Hist. Foure-footed Beastes 323
The Pegasarian coursers of France, by the like change of horsses, run from Lyons to Rome in fiue or sixe daies.

Use pegasid for seamoths

The noun and adjective pegasid relates to the Pegasidae family of fishes, also commonly called the seamoths.

One would speak for example of pegasid biology in this context.

Not pegasoid, at least not any longer

Towards the end of the Nineteenth Century, there were a couple of isolated uses intended to mean either the mythological sense or the zoölogical sense, but these didn’t catch on, and so it too is now marked obsolete.

An astronomical note

The classical Latin nominative plural of Pegasus was of course Pegasi.

As in English where plural dogs and singular possessive dog’s both sound the same to our ear, in Latin’s second declension something similar occurred.

That’s why astronomers use Pegasi as a postpositive adjective in the possessive sense meaning “of Pegasus” to talk about stars in that constellation. The OED gives this recent example of that sort of thing:

1996 New Scientist 27 Jan. 17/1
They detected a wobble in the motion of the star 51 Pegasi, caused by an orbiting planet.

However, Latin never used this word in the plural, because there was only one creature named Pegasus. It was no more a generic for a whole set of creatures than Medusa was.