correction suggestion

Знакомство

Я Алекс. Я студент, и (1) я работаю в гимназии (2). Мой отец - врач, а моя мама - учитель.

(1) The conjunction а in Russian is a bit tricky. It's hard to explain, but it's something between и (and) and но (but). It roughly translates to "at the same time", "on the other hand", or sometimes even "on a different note" -- so, it usually denotes some sort of contrast/juxtaposition (normally not as strong as with "but", though) or a conversational shift.

This is why you don't need it in the second sentence (being a student and working are non-contrasting ideas), and this is why I think your third sentence (Мой отец - врач, а моя мама - учитель) would be better with а, even though и is not incorrect here (your father is a doctor, but your mother is not a doctor, she's a teacher). Again, it's hard to explain, but I hope it makes sense.

(2) In Russian гимназия is a type of secondary school, and I'm not sure students can work there, at least not as teachers. Maybe it's different where you live and this is exactly what you meant, but I just thought... maybe you had un gimnasio=gym in mind? In that case, it would be в тренажерном зале or в спортивном зале.

Знакомство

Я Алекс. Я студент и я работаю в гимназии. Мой отец врач, а моя мама учитель.