On paper both are independent entities. But the reality is a bit different. The EU parliament holds the key to the budget. Than there is the EU's common security and defense policy that governs the European defense union (EDU) which is ironically headed by EUs top diplomat who will be Kaja Kallas.

The EDU has a lot of say in how EU missions are conducted. It has sway over defense production. There is significant overlap between the edu and nato. (Will add more after get to the laptop. Hate typing on the phone).