There are a couple of autism-supportive groups who get table/tent spaces at our local pride. Although there are some church booths there, too, it's not exactly what you'd call a sacred space. The whole religion thing is rendered irrelevant. Super friendly, and the people are there to do nothing but connect people to other people, and organizations.
If you want to get really friendly, and have the chutzpa to be pro-active, volunteer at one.