I vote a solid (and unpopular) no.  I have been at the park with two kids  (from ages 1 - 26).  We brought a stroller on one day and one day only (the year we had a 1 year old and a 6 year old).  I almost ditched it in the park.  It made everything so much more difficult.  Maneuvering, parking, people wanting in, people wanting out - misery. 
Before we had the younger one, we had been three times with my older daughter (she was 2, 3 and 4).  She walked most of every day.  We gave her a bag of skittles - as long as she was walking, she could be eating. We carried her every now and then - mostly after parades and fireworks.  The younger one had more energy than all of us - she never complained about the walking. 
We also do not bring a lot of stuff into the parks.  Back then - a small backpack with a change of clothes for the kids and a diaper or two.  That's it.  I never feel that we are going off into an abyss - so I can buy something if I need it. Which - thinking back - I never needed to do except for food.  I am not lugging lunch into the park.  That is not my definition of a vacation.
Also - I think you experience the parks more with your kids if you are walking along, holding their hand and talking to them.  Pushing them along in a stroller while they suck on a sippy cup in a crowd of people - I don't get.