As a work-from-home dad with a very flexible schedule, I spend quite a bit of time at the park during work hours. My experience is closer to yours than most others in this thread. I find the moms/nannies/grandparents quite welcoming and friendly. Occasionally a topic of conversation will come up that may be a bit awkward, but otherwise I don't feel that I'm treated much differently. I certainly have never perceived that anyone felt threatened by my presence.
I'm not sure what would account for the difference. I live just a few miles from FB headquarters and am in the same demographic as OP.