In the park there are 4 zones. To start off only one is unlocked; this one looks like a traditional park. You can have one of your pets dig by clicking the shovel in the bottom left screen; then clicking a place on the ground. You can dig up precious gems, or items to complete a quest that will lead you to unlock the next zone.
Digging costs 5 energy points; but you can buy more with facebook credits, ask your friends to give you some energy, or simply wait to recharge these points. The energy bar fills up at 25 points.
In the first park zone you must dig to find building materials to build a trophy room, when finish digging up all the necessary materials you will unlock the trophy room in your house and the next zone in the park.
Once you reach the next zone, which looks like a desert, you will have the opportunity to dig up T-rex bones. After you complete this quest you will revive a T-rex skeleton for your house and access to the third zone.
In the third zone, which looks like a farm