Undo the parenting that you did with CTRL + P. It's handy, but it's limited.
Then select the camera and go to constraints. Select constrain "Follow path".
Then choose the curve as target, in the constraints menu.
Now you can select the curve and go to the "data" tab(the one that looks like a curve).
Under "Path animation" you can animate how fast and far the camera will move on the curve.
If you select the camera, you can animate the rotation as a normal animation(You can reset the position of the camera with "alt + g" to snap it to the curve).