In my experience as an angel investor this is authority remains with the board.
In your case, the reason they may be asking for budget approval is they don't know any better OR because of the make up of the board. Having 2 founders on a board of 3 is unusual. With a board of 3 usually the founders get one member (usually the CEO) the investors get one member and they get to agree on the third.
The reason for all this is that investors want founder's access to wealth to be aligned with their own - meaning not till they get an exit. They don't want you to be able to give yourself high salaries, take the company into debt that will dilute their ROI, have your company retreat in a fancy resort in Monaco and then not have cash to make the company grow fast enough to achieve an exit.
I have seen where the holders of preferred shares get veto power over some of that stuff - like founder salaries beyond a certain amount, company debt. Though I have never seen anything written about resorts in Monaco [grin].