Philana says her people stop aging, yet Liam Sullivan(Parmen) was 45 years of age, and he clearly looks even older than that.
Leonard Nimoy's double is just visible as Mr. Spock loudly dances "flamenco" dangerously beside Captain Kirk's face. When the dance finishes, it is obvious that the dancer's boots have higher heels than the ones Spock is wearing.
Spock's ears are quite obviously fake prosthetic ears when he and Kirk wear the laurel wreaths on their heads towards the end of this episode. His left ear in particular has been bent to prevent the wreath slipping down any further and because of this it looks very abnormal and fake.
The Platonians repeatedly mention "Utopia". It is impossible for them to know the term, as it was coined by Thomas More in 1518 and they left Greece centuries before he was born.
In his Captain's Log, Kirk states that the Platonians' "planet went nova." Planets don't go nova, only stars.
During the performance by Uhura, Nurse Chapel, Kirk and Spock, one of the Platonians refers to "Cupid". Cupid is a Roman deity; in Greek mythology the name of the deity is "Eros".
Parmen refers to the "pièce de resistance," a French term which did not exist when the Platonians visited Earth.
When Spock sits next to Alexander and begins laughing, actor Michael Dunn breaks character and begins laughing with him. (Uou can see he's trying unsuccessfully to restrain himself.)
If the Platonians left Earth at the time of Plato, they could not know about "Tweedledee" and "Trweedledum".