Some people ( Footiefan in particular but also all those insisting someone has to be "a twat" and as it isn't OP it must be the dad) are blowing this tiny non event completely out of proportion.
This tiny interaction worried the OP because she has anxiety. This is understandable in light of her own personal mental health. It's important for the OP to realise that objectively the interaction is not significant, and only her anxiety is making it seem so.
It is not at all appropriate to label the other party a twat and imply that his child is going to grow up disfunctional and unable to accept authority or that the dad and child should be avoided at all costs because they are dreadful human beings. FFS.
That reaction is even more disproportionate and inappropriate than the OP's anxiety driven response!
Nobody is a twat in this scenario. The dad's response may well have been completely appropriate under the circumstances because nobody reading knows whether there's a back story (sensory issues, anxiety on the part of father or son - just because one party in the interaction has mental health issues doesn't mean nobody else involved is neurodiverse or dealing with similar issues - they very likely do have any issuesn't, but as we're all rightly understanding the OP's issue why not keep an open mind). After all the father's primary responsibility is to his child, for whom the coat issue might be a big deal, or for whom learning to say no might be a big deal - some children are overly eager to please despite being very uncomfortable and this is sometimes even more of a problem than being deviant.
Whatever was going in nobody on this thread knows and the reality is that the mentally healthy way to respond to the interaction is by filing it under "very trivial, brief interaction requiring no further thought" and carry on.
Worrying yourself sick is an inappropriate response caused by anxiety, but the aggressive determination to label the dad and other child as twat and brat and write them off as people to avoid at all costs is even more mentally unhealthy! Labelling anyone you have a slightly unsatisfactory interaction with as (effectively) an enemy or terrible human being is not the way to retain calm and mentally balanced!