I would start by teeing in a pressure gauge in the park hose and see if pressure is keeping up when you open faucets. If it's not, there is a restriction in the park plumbing. If the pressure is staying up, I would try filling the fresh water tank and see if the flow is better when using the RV pump. If it's good, no more problem. If not, it could be bad pump pressure. I had low flow when using the pump and I had to turn the pump pressure up. If flow is still bad, disconnect the pump discharge connector and connect the gauge to the discharge tube with a 1/2" pipe fitting. Then you can use the park water again and see if the pressure is staying up in the RV system while faucets are open. All this will greatly narrow down where the restriction is. Craig
On second thought, I'd use two gauges. Keep one teed in where the park water enters the RV and one on the pump discharge tubing. I think it would be good to see the pressure at both those points while faucets are open. If pressure is good where the water enters but low at the pump discharge gauge, the problem is most likely the check valve at the park connection cuz nothing else should be between those two points. If pressure stays up at the pump discharge gauge and flow is still bad, there's a restriction between that point and the faucets. Good luck.