If you're just going for a quick walk around the park pocketing a compact will be fine but for those who will be taking their camera and interchangeable lenses along, pack your telephoto lens for shots of dog walkers, animals and long shots of the park with your town in the background. For close up work with flowers and leaves pocket your macro lens.
An overcast day, when the light is gently diffused, is the perfect time to capture flowers and foliage. They look even better after a light rain shower as the vibrant greens will jump right out of your frame. If you spot a squirrel while looking through the trees and bushes keep your distance and use a telephoto lens to fill the frame with the cute animal. For those of you who have streams, ponds or even lakes in the park you're visiting why not have a go at duck / swan photography. In most locations the ducks are used to people so getting close to them shouldn't be a problem.
If it's a fine day there should be plenty of people for you to snap a few candids of as they pass by. Try shooting from the hip or if someone such as a park warden tidying up really catches your eye, remember to ask them if they mind you taking their photo before you snap away.
Paths and lines of flowerbeds can be used to guide the eye through the image while repetitive patterns such as lines of trees, street lighting and fencing can add symmetrical interest to your shots.
If your park has wooded areas you could have a go at capturing beams of light bursting through the trees but this usually means you'll need an early start and shooting on a misty morning is a must.
Don't forget that statues, monuments and water features make great photographic subjects and most parks will have one, if not all of these on display somewhere. Try looking for glimpses of buildings in your town through the leaves on the trees or gaps in the hedges. The contrast of green, or even the golden colours of Autumn later on in the year against the glass and concrete of the town can work extremely well. Then, when you leave the park, find a hill to climb and photograph the patchwork of greens that break up the grey lines of the town.