I don't want to burden my parents any more.

Literal

Parents [to-に] this-more-than burden [object-を] don't-want-to-impose.

これ以上 (more than this / any further) sets a limit. 負担をかける (to impose a burden) is a set expression: 負担 (burden) + をかける (to impose on). ~たくない is the negative of ~たい (want to).