First and foremost, I'm sorry you're going through this. It sounds really horrible. Social anxiety can be really awful and I suspect a lot more people will relate to this sort of problem than you might expect.
In the short term, you might absolutely want to embrace any chance for 'drop and run' type social experiences for your daughter and reduce the pressure on yourself. Especially as she gets older, you won't be holding her back socially by being reluctant to meet up with other parents extensively. You might have tried this already but perhaps structured activities might help you to feel less anxious where possible - personally I feel way less awkward around other parents when my DD is doing some sort of class or craft activity, and I have the coping mechanism of taking on a lead role with the kids (eg chasing them around the soft play or busying myself with running a game) because I actually feel more comfortable doing this than chatting to the other parents at times.
There is a big but coming though...
In the longer term, avoiding stuff fuels anxiety. That is a basic and unavoidable truth. If you stay away from other parents and remain trapped in the way you are thinking and feeling about yourself, that could be really damaging for you.
The NHS often offers pretty good therapy for social anxiety via Talking Therapies. It is quite structured and practical. I know that CBT and NHS therapy in general can have a bad rep but I wouldn't rule it out at all. It does genuinely help a lot of people.
There are also some lovely self-help books for social anxiety and it might be useful to browse the free extracts / intros on Amazon to find one you like the tone of. Nothing worse than a book written in a style you find patronising and annoying. Some have audiobook versions - again, sample before buying to check you like the narrator.
Even without all that, I really hope you can be compassionate to yourself. We get anxious for all sorts of reasons and it tends to make total sense in the context of our life experiences. It also isn't fair to catastrophise about what your anxiety will mean for your DD - I'm sure you are a fab parent in loads of ways and no one is the 'perfect parent'. She is still very young with a lot of opportunities ahead. You are doing the best you can right now.
If you feel you don't fit with the parents around you or in your friend's area, that might partly be because you are assuming the worst about yourself and are too much in your own head, judging yourself and worrying about what others think, to make connections. Having some strategies to cope with your anxiety and push through could help you realise that your people are there amongst these parents already.
On the other hand, anxiety isn't always just 'in our own head' - perhaps you genuinely haven't found people you click with. If this is the case I urge you to think about your interests and try to meet more people through activities you actually like. Some of us are just more niche people than others and more niche in our interests but there are often loads in the same boat and online communities can help you to find people in your area.
And as someone who went to Oxbridge myself, I can assure you that there isn't just one 'Oxbridge type'. I knew a lot of people at uni who I found obnoxious and up themselves for sure but many of them turned out to be interesting and lovely once I knew them well enough to get behind the bullshit. I managed to find other people at uni who suited me and there were all sorts there. I also don't think an Oxbridge education or indeed any fancy education makes someone any smarter or more interesting than anyone else. Most of smartest and most interesting people I've met in my life never went to uni. Not everyone is shallow and judgemental and status-obsessed (although sadly I know some are).
Wish you loads of luck.