Honestly, the best way I've found to move a heavy park bench across a lawn without scratching the grass is to give the bench a temporary path rather than dragging it. Heavy benches concentrate all that weight on four small legs, so even a tiny slide can tear the turf and leave brown streaks.
Start by measuring your route and clearing any sprinkler heads, roots, or rocks. Grab two or three sheets of 3/4-inch plywood—OSB works too, but plywood is smoother—and lay them end to end like a plywood boardwalk. With a helper, lift the bench and set it on the first sheet. Then slide or walk it from one sheet to the next while your helper moves the rear sheet to the front. This leapfrog method spreads the weight and keeps the legs off the grass the entire time. Lift the bench; don't slide it across the board edges, or a leg can catch and dig in.
If you're moving it solo, a heavy-duty furniture dolly with large pneumatic tires is your best friend. Put a piece of plywood on the dolly first so the bench legs can't punch through the frame. Lift one end of the bench at a time, slide the dolly under the center, and strap it down tightly. Roll slowly and avoid sharp turns. The wide tires float over the lawn instead of rutting it, but if the ground is wet or soft, wait a day or two—wet grass tears much more easily.
For very heavy cast-iron or steel benches, use the roller trick. Lay two or three thick PVC pipes or wooden dowels on the ground, place a piece of plywood on top, and set the bench on that. Push the bench forward, and as the rear roller pops out, move it to the front. It sounds old-fashioned, but it works surprisingly well and spreads the load over a larger area.
Whatever you do, don't drag the bench legs across the turf, even for a couple of feet. That's how you get skid marks that take weeks to grow back. Protect the grass first, then move the bench—it takes a little longer, but your lawn will look like nothing ever happened.