Thanks all for your thoughts and responses. Viewing the situation this morning with a clearer head I agree that the playground was a red herring and actually the real issues are the other things I mentioned.
To clarify- I do think DH is a good parent on the whole- he isn't the parent who will organise play dates/exciting activities/trips (not that we can do those anyway atm) but he will take her out for long walks in good weather on his day off. He can also be very lazy though with her- sitting on phone scrolling whilst she gets bored/fussy or trying to get her to speak to family members on video call which she absolutely hates on the whole and has no interest in doing. During lockdown it was a necessary evil but now she is seeing family members again I don't see the point. He does (and always has done) nappies, getting her dressed, getting her fed etc. (Although in my mind that is just the BASIC expectations of any parent, so no gold medals to anyone who does that). DH also does all the cooking for he and I and has done ever since DD was born.
A few things other posters said have struck a chord about dictating playgrounds sound controlling. On reflection it does and it is. I have an anxiety disorder and so one (unhealthy way) I can try to manage it can be by trying to control everything. I am fully aware that that must be exhausting to live with when I am in that headspace. I think DH and I need to discuss some ways we can speak about that if he genuinely feels that's what I am doing.
RE: swearing and 'shut up'- the first time it happened (I can't remember exactly when) I was incredibly upset. Mainly because it had happened in front of our DD. I spoke to him at length about why I found it unforgiveable and inexcusable e.g. the way he treats me in front of her is how she will allow herself to be treated by future partners and I would never EVER want her to be with someone who would speak to her like that.
It has happened once more I think in front of her since, and 2 times more not in front of her. Timeline for all this is probably about a year.
I have spoken to DH before about 'sulking' and silent treatment- and a refusal to discuss issues that have arisen. I've told him I find it childish and immature. He has acknowledged he isn't good at discussing his feelings etc. he said he would work on it. There has been improvement in that silent treatments aren't happening so much anymore but instead they seem to have been replaced by the occasional sudden outburst of swearing and telling me to 'shut up'.
Another PP said he clearly views me as hyper critical- I agree. He is naturally lazy- so are his parents. I would say I can be lazy too but equally my DD motivates me to be better. We all have days we want to just sit in front of netflix or scroll on our phones- however 2yos don't allow for that (unless they're napping), so I just get on with it on a day like that. DH definitely feels the burden of parenting and views it as such- although he also gets huge amounts of enjoyment from our DD and loves to talk about her after she's in bed etc. He will always complain of being exhausted and tired- yet he won't go to bed before midnight, even knowing he is up with her at 7 on his weekend 'morning'. So he whinges but won't change it.
I will be wanting a discussion today about last night and as another PP has put- I will be making the swearing at me a red line from this point onwards. If it happens again then he will need to leave.
For those saying LTB- I don't think we are there yet. But rest assured I would be the one staying in the house with DD and not him.
Thanks all for your opinions.