As a photography enthusiast from Budapest, Hungary, I have been wondering for years which are the best spots to take pictures in the city. It has been an adventure to uncover most of them and I believe it would be quite difficult for someone who is not living here to discover the hidden gems.
The city has many architectural wonders that can be perfect for any artistic project, starting from the Parliament which is known by everyone and Buda's Castle. I would personally recommend capturing the views from Gellért-Hill or the Széchenyi Chain Bridge.

I have been to Budapest once, when I was young. I remember how impressed I was of its architecture. When I will visit it again, I will go to the locations that you have mentioned and try to snap some photos.