Here's where I technically disagree. The FCC definition of FB2 is mobile relay station. The exact definition is located in Part 90.7.
By that definition, if it automatically repeats a signal that originated on the transmit frequency of a a mobile station...its a mobile relay regardless of whether or not it is repeating on the same frequency or a different frequency.
That being said, I wouldn't configure a parrot repeater to receive and transmit on the same frequency if it could be avoided. The reason being, you have four types of user experiences in this situation assuming everyone is within range of the repeater. The originator of the message who hears what they just said, those who hear the real-time message and then the parroted message, those who hear a bunch of noise and then the parroted message, and then those who just hear the parroted message. If a parrot repeater is instead configured to receive on one frequency and then transmit on another (like a conventional repeater or split-base) you only have two types of user experiences. The originator who hears what they said and then everyone else who only hears the parroted message which can greatly simplify and lessen confusion.