Well if you have a DSLR and don't want to bring it to certain parks or places, you've got a few different directions you can go.
1. You can get a mirrorless system camera - nearly the identical quality and low light ability as the DSLR, in a more compact and lighter package, and not terribly expensive. You still might be a little scared throwing it around in the bottom of a baby bag or sitting it on the sand at the beach, but it's so much more portable than a DSLR that you might be willing to bring it along more in the parks and rides.
2. You can get a decent P&S ultrazoom - something very versatile that can shoot from wide to extreme telephoto - image quality and low light won't approach DSLR levels, but convenience and versatility will be very high in a fairly inexpensive package...and more portable to boot. Some 'travel' zooms can even give you decent reach and fit in a pocket. Still not made for abuse, but won't have you wanting to post armed guards around it because it's inexpensive enough compared to a DSLR.
3. You can go for pocketable P&S cameras with some control - if you're a DSLR used or pro, and like to have control over your shots, you may prefer to still have A & S priority and Manual modes, capability to shoot RAW, maybe a nice fast lens...look into the high end pocket cams like Panasonic LX5, Canon S95, Samsung TL500, or Olympus ZX1. Pocketable or close to it, but still lots of nice higher end features. Image quality better than other P&S, but still below mirrorless and DSLR.
4. Go for the bulletproof all-out fun cameras - waterproof, shockproof, etc. Basic P&S stuff inside, a good photographer can still get good photos from them, you accept they're not going to match a DSLR, but they can be thrown in a pocket, left in the bottom of the baby bag swimming in crumbs and baby powder, can be used on wet rides or in the wave pool, can lie in the sand at the beach without a care, can be dropped on the ground...made to be used and abused and still get shots when you otherwise wouldn't want to be carrying a camera. Canon, Panasonic, and Sony all make nice higher-end ones with some cool features, Fuji, Casio, Samsung, and others make cheaper ones that might not take as much of the bumps and drops but cost half as much.