"No cops at pride" is not about the prejudices of individual cops, be they fictional future shapeshifters from half a galaxy away, or real police here and now. There are LGBTQ+ cops out there.

The issue is the fact that cops enforce the law regardless of how just the law might be. Odo was the chief of security aboard Terek Nor while it was under Cardassian control, and while in that role rushed three innocent Bajoran workers to execution so he could maintain order aboard the station.

Even once the station became Bajoran owned and Starfleet operated, Odo was still willing to conduct illegal surveillance, lock people in the detention facility on trumped up charges, and impose a strict curfew. Personally I don't think it would be too much of a stretch to assume that Odo would be willing to lock up people participating in a Pride event for no other reason than that he was told to do so, and they were causing a minor disruption on the Promenade.