Is it possible to walk along the Thames, all the way past the Houses of Parliament? Or would one have to leave the river and go around the Palace? Thanks
No. You would have to cross the road and walk round the corner to then walk the length of the building, where you can then continue riverside walking.
The riverside terrace at Parliament is members only. You have to go around it from Westminster Bridge (where I was born at the General Lying In, Waterloo) to the park the other side via Parliament Square.