I think the number 1 rule of riding a road bike fast is to give pedestrians the right of way. But if you live in a huge city and there are only a few places cyclists can ride and there are points of friction between riders and park users, this primarily is a policy and infrastructure failure.
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It's a park and recreational (fast) road cycling should be encouraged, but as you say pedestrians do need to get priority.
BTW, It's the through traffic that needs banning.