Legal Question in Employment Law in Texas

Was your part time position covered by a contract? If so, the contract will govern when the school can fill your position. If you are/were an at-will employee then your job carries very little protection. If you accepted the new position then the school was likely free to fill your old position. If you worked for a public school then you may have some other means to preserve your job.

It's hard to say what rights you may have without knowing more about the terms of your employment and details of the situation at hand. Your best option is to speak with a local employment lawyer about your situation.

Adam is on point with his assessment. The contract governs or if you are part of a union then your union contract may provide the understanding of what your former employer could and could not due. If you are simply an employee at will then your former employer may be well within its right to have done what it did; however this is just a general assessment based on limited facts and information. You would be best served talking to a local employment law attorney as he/she would give you more specific advice based on facts that you would provide.