To be honest, the only type of puncture you're more likely to suffer as a result of thinner tyres are pinch punctures. I don't really know your bike knowledge, but if you're unfamiliar with them, pinch punctures happen as a result of the tyre bottoming out between the ground and the rim due to insufficient air pressure or riding like an eejit. The actual size of the tyre makes no difference as to any other type of puncture.
Tyre wear again, isn't really affected by the size of the actual tyre, but more the rubber compound and tread type.
For what it's worth, I do prefer bigger tyres because as above, I do occasionally ride like an eejit and probably beyond what my bike was designed for. A bigger tyre can be run at lower pressures thus giving a smoother ride but they can and do clog up the mudguards quicker. But if riding in the dry, that's not really an issue.
What I would say is keep your bike as it is for now. Ride it a bit longer and see how you get on with it at the park. If you feel that you'd like a softer ride, let a bit of air out. If that causes the tyre to squirm too much or the rim takes a knock, then consider fitting fatter rubber.
I used to ride with a bike club. All members used to go out on skinny 23c - 25c tyred road bikes. One day I suggested that instead of coming home via a very boring main road, we ride along the canal path. Some worried faces followed as they were concerned that their bikes would be slipping and sliding all over the gravel surface and taking a battering. We did it, no one died. A short time later I had to leave the club due to family commitments but I heard that the canal run was made a regular thing.
Bikes are tough old dogs.