Am I respondsable for paying the entire amount of the estimate that they sent to me?

They have sent you a bill. You can pay it or not pay it. You can try to negotiate. You can file a lawsuit, they can file a lawsuit. These are some but not all of the possibilities.

If this happens in WA, your driver's license can be suspended for driving without insurance. (It probably the same in all other states.)

A person backing out of a parking space generally has a greater duty of care than a person driving forward. Which driver fails the driver's duty of care in your particular case cannot be determined merely from your post.

A lack of insurance is not a factor in determining fault.

The $824 demanded from the other side may or may not be reasonable. You can negotiate with the other side on the claimed damages.

What you may have to look out for is the other driver claiming personal injuries from the impact. A personal injury claim would be much more than $824.

You may want to check if you are required to file a collision report with the State Patrol. Information is available here: http://www.wsp.wa.gov/publications/collision.htm .

Not buying insurance can turn out to be much more expensive than buying at least the minimum required insurance.

You may want to review your facts with an attorney.