Dirt is one thing but I'd avoid mulch at the park. Commercial mulch or wood chips can contain all kinds of wood from yucky chemically sources or poisonous woods. Every year on my equestrian lists there is someone whose horse gets poisoned from them having used commercial sawdust or woodchips as bedding. Plus mulch is designed to hold moisture so it is more likely to be teeming with mold spores than plain old dirt.
I was always one of those clean kids; though I would get terribly messy, playing outside I never got dirty. I also didnt like a lot of rough games and might have told an adult that I was afraid of getting dirty while I looked longingly at the other kids wishing they'd play something else, but I don't think not wanting to get dirty ever really kept me from doing anything I wanted to do, just if I did get dirty I wanted to get clean again fairly quickly (I'm still like that actually, and I'm still miss messy). But still I know what Johub means..I had cousins whose mom would SCREAM at them when they came to visit if they ran around and got a little sweaty, if they ever got dirty she would have blown a major fuse the one time they got a little mud on their legs (and I mean just a few spatters), she made them take a bath at our house.
I was always messy but clean, one brother was as neat as a pin but would get dirty, and my other brother was pigpen..always a total disaster within minutes of having had a bath so it had nothing to with our parents, it was just how we were. Because of my dislike for dirt though I feel like I need to be careful not to influence my son too much..I think I wipe his face too often because I don't want to get the mud or dirt on me. However, since he is a big water baby, at the moment he thinks washing his hands or wiping his face is fun so no conflict yet. Hopefully there won't be an issue and he'll like getting dirty and getting clean again
As a preschool teacher I had kids who wouldn't participate in messy stuff because they didn't want to be dirty, I found that if there was a way for them to clean up periodically during the activity (not making them wait til after it was over) they would often choose to participate. For kids that are really phobic about dirt its often a self image problem and doing mirror excercises can help them be more confident about it but for most kids its just a sensory preference and doesn't mean they aren't having fun.
I was glad to read this thread by the way, I always feel weird because even a lot of the crunchy mom's around here use shopping cart covers and take toys away if they fell on the floor. We got a shopping cart cover as a baby gift and I ended up returning it because I didn't see the point of having to carry it every time we went out.