Since we don't have the agreement you signed, you need to have a lawyer review it. Otherwise, you take your chances and no one here can tell you what your former employer will or will not do. Does your family have a contract with the valet company? If so, that is another issue. If you are going to start a business, it is always wise to start with a good business lawyer and a good CPA.
Mr. Riddle's answer is correct. There are a number of material issues that your lawyer would need to consider which were not addressed in your post. From the basic facts outlines, I am hopeful that a creative solution exists. An experienced lawyer would probably be able to identify the safest means to achieve your goals after an initial consultation, review of the contract at issue and some basic research. That said, there is never a guarantee that the other party will not take legal action against you. Before you take any steps, take the time to sit down with a lawyer of your choosing and discuss the situation at length. This might cost you some money up front, but it could save you thousands down the road.
Best of luck.
The above is not a full and complete answer in that it is based on the limited facts known at the time of the answer. With that in mind, the above is intended as a starting point, not to be construed as legal advice. If after reviewing the answer you still have additional questions, please feel free to contact me through my office so we may discuss the matter more fully. I can be reached through our website at foxandsmith.com
I like the idea of finding a creative solution. For one thing, the people you want to serve are your family and their restaurant. I infer that the first valet company was an independent company. Would the scope of the non-compete agreement cover a situation where you worked for the family restaurant (as its employee, not a separate valet company) and your duties were the valet services? That may be something to have a lawyer look into. Good luck.