So this one is tricky. If you said "I'm wearing a black and white armband" I will understand that you have One armband. Because you said "I'm wearing Both a....." I'll understand that you are wearing Two armbands: one black, one white. However "I'm wearing a black and a white armband" sounds natural. When it comes to people, especially a set of parents. You don't use Both in this way because the assumption is that you're talking about two people. Here Both should emphasize something special or specific, such as "My parents are -both- Mexican and Black".

In both sentences, the second "a" (the one you have in parentheses) is required. In "a black and a white" and "a black and a Mexican," the sentences mean you are talking about two armbands/parents. Best wishes with your studies.