Newton's third law and the phrase "unbalanced forces" need context. The third law states the following: Whenever object A exerts a force on object B, object B exerts an equal and opposite force on object A. Notice that there are two different objects here, and each force is applied on a different one. Put in a different way, the law says that if you sum all the forces on all the objects in the universe you get zero, but it doesn't say that if you sum all the forces on each object you get zero.
Unbalanced forces usually refers to a single object. What it means is that the sum of forces on something is not zero. For example, if you're sitting still on a chair the forces on you are balanced: your weigth points downwards, the normal reaction from the chair points upwards, and they cancel. Both of them are applied on you; they are not an action-reaction pair, which are the concern of the third law. If you now imagine you're in free fall in a vacuum (please don't put yourself in free fall for the sake of experiment), there are unbalanced forces: weight pulls you down but there's nothing pushing up to compensate, and so you accelerate downwards. Newton's third law is not violated: the Earth is pulling you down, and you're pulling the Earth up with the same force.
The rock is not a great example because it being hard to move doesn't have a lot to do with this. One of the reasons it's hard to move is friction, which is roughly proportional to weight (in most circumstances). But even if you placed the rock on ice so it slipped, it would still be hard to push because of Newton's second law: since the rock has a large mass, it acquires a low acceleration. The third law is irrelevant because it concerns the force that the rock exerts on you when you push it. If you try to push while on ice skates, you'll just go flying backwards.