When I parent an object to follow a path and animate it, I only get a linear mode animation and changing that in Grapth Editor to bezier doesn't work! How to do that?
When you parent an object to a curve with Follow Path to control its motion,
In the curve editor you'll be presented with a straight line for the evaluation time with no keyframes or any way to control the interpolation. To fix that press N to access the properties for the curve, look for the modifiers section and erase the generator for the curve.
Now you can manually animate the motion of the object by animating the evaluation time on the curve using keyframes, those keyframes will default to bezier interpolation.