The Pride parade and the Pride movement evolved out of a not too distant past when individuals who identified as gay, or existed outside societally prescribed norms, faced hate and violence on par with that faced by many other historically persecuted minority groups.
While it is certainly more of a celebration now than it was during it’s earlier activist incarnations, the Pride parade still serves an important function: That of publicly affirming, for those who know or remember what it’s like to live in fear and face discrimination, the right to openly live their truth—safely and without consequence. It’s an opportunity for society to show that it is evolving, that it has evolved away from a bigoted and judgmental past, and that the status quo now includes support for those who were marginalized just years ago.
The problem with the “straight parade” seems to be the same as the problem with the “all lives matter” thing. It’s not that the premise is inherently false, but that it doesn’t require public communal support in the same way. People don’t live in fear of exposing themselves as straight—we have not ever, to my knowledge, been historically persecuted for being straight. In America we have not ever been historically persecuted for being straight and white either. To say “all lives matter” is to say an obvious thing. To say “Black Lives Matter” is to say “I am on the side of progress and change in this country”. To say “straight pride” is to say an (odd but acceptable) obvious thing. Ok, you’re straight in a heteronormative culture. Cool. To say “Gay Pride” is to say “I am on the side of progress and change in this country”, I can publicly show support for a historically marginalized group without fear of repercussion.
That’s the point of the Pride Parade. It’s important because collectively we need to continue affirming the rights of all different sorts of people, until the fabric of our culture really does absorb that all different sorts of people should, in a sufficiently advanced society, share the same rights and protection.