NatLuc, you're not alone! I'm a transgender parkrunner to. I recently got a story about me in the Australian parkrun newsletter, and I'm quite possibly the trans woman the OP was complaining about that started this whole discussion.
One thing you need to realise though is that mumsnet favours a very specific set of beliefs around transgender people and our identities. It does not reflect the larger experience you'll have at parkrun.
In my case, since the article went live, I have been drowned by messages of support, including the various women's running groups I'm a member of. Of course, there have been negative voices too, but they are small in number, and tend to be restricted to specific spaces where those beliefs are welcome.
And in terms of advantage, before my transition, as a 41 year old, running against males, my PB was 19:41, which means that I ran 70% of the male world record for my age.
Since transitioning, hormones have drastically reduced my athletic ability. My best time now is 23:00, which has me running at about 67% of the women's world record for my age.
That is to say, even though my times have dropped, my performance within my gender category has stayed roughly the same. I haven't found any advantages, and in fact, I'm performing slightly worse against women than I did against men.
And the research shows that for transitioning runners at least, my experience is typical. As long as our training stays constant, even though our times drop, our age grading stays roughly where it was.
That's the thing that I find frustrating in all of this. So many of these comments are based on assumptions of advantage, and for running at least, they simply don't exist. We don't see trans women runners dominating their sports, we don't see large changes in performance, with ladies running better against women than they did against men. The (admittedly limited) research in this field demonstrates the same. If transitioned trans women had an advantage in running, we'd have seen it by now