"They then rounded us up into a large room and weeded us all out by checking everyone's IDs.
The officers are also said to have arrested two other "foreign" women at the Playboy Music Fest who were "special guests" from Paraguay and Peru.
Elaine, Brethenoux and Celik are the only three to have been named, and were detained because they were deemed to be working while only on a tourist visa.
They were then placed in a holding cell cage for 24 hours.
But Elaine said the accusation was nonsense.
"We do not get paid for these events," she said, "so we were wrongfully taken because they abused their authority and thought they could shake down Playboy Mexico by taking us."
Celik described the event as "a horrible moment in my life that I have never had before."
Elaine credited Playboy Mexico with "pulling strings" to get them freed. It has since described the arrests as an "abuse of authority" and a "misunderstanding".
In the statement, the publication stressed their models were not working but were invited to "mingle with the audience" at the event.
A spokesman said: "We regret the treatment the women received at the hand of the authorities in that city and thank you very much for the understanding."
After being freed, Brethenoux posted an Instagram photo of herself and the other bunnies pouting, with their hands crossed at the wrists.
"Even after jail in Merida nothing can stop us!" she wrote. "Bunnies warriors."
The women have yet to get their passports back, but plan to meet with immigration authorities soon.