Is it permissible to continue studying at a mixed-gender university, despite the corrupting influences of mixing therein, or should one search for another non-mixed and academically suitable university, even if that necessitates traveling abroad, given its conflict with family circumstances?

Studying in a mixed-gender university is a grave evil (shar kabir) and is not permissible except in cases of necessity (dharurah) or extreme need (hajja shadidah).

If the conditions for permissibility are met, and your parents need someone to serve them, you can arrange with your siblings to take turns serving them, or delegate a trustworthy person to do so, because serving parents falls under the obligatory financial support (nafaqah) due to them.

If you are unable to travel due to your parents' circumstances, then you must be modest, lower your gaze, guard your chastity, minimize evil as much as possible, choose righteous company, increase in voluntary fasting and remembrance (dhikr), and hasten to marry if it is facilitated, for it is more effective in lowering the gaze and guarding chastity.