As soon as you start working in your second job, you will be violating your first contract. If you refuse to start working in your second job, you will be violating the second contract. You were very careless.
Not much of a legal question, but the question is how to get out of the mess you created at the lowest possible cost. I would suggest that you go as soon as possible to the second company (the one with the part time contract), tell them that you are very sorry but you didn't read your first contract properly, and that you cannot start working for them. If you seem suitably sorry there's a chance that they will tear up your contract and send you home, with no more damage than a big red "DO NOT HIRE" in you records.
If things don't go nicely (and you don't have any legal right to expect them to be nice about it), you go and get a lawyer.