Difference between a parent class and super class

I'd say it's the same.

You might want to differentiate between a direct and indirect parent or super class, but I guess the two terms are not clear enough on this, either. So if this is what you are trying to express, better be explicit.

Also, many programming languages have the "super" keyword used to refer to the (single) direct parent class. But even there, if you call a "super" method and the direct parent does not implement it, it also bubbles up.

This is more of a terminology difference, the idea of parent and child classes or super and subclasses. It seems to depend on programming language experience and application domain as to which one you use as well as when you first began getting into Object Oriented Programming.

In both cases there is a class, the parent class or super class or base class, from which is derived other classes, the child class or subclass. The child class or subclass extends the parent class or super class by adding some capability to the existing capability of the class being extended.

super() is how the parent or super class constructor for a Java class is invoked in a derived class.

There was a fair amount of churn in the terminology during the first years of object oriented programming as various people worked in the area and published papers and books and developed Object Oriented Languages. It was all quite new and exciting and people were trying to decide the proper vocabulary to use so they were trying out various words and phrases to express Object Oriented concepts.

And with a number of Object Oriented Programming languages that have been developed and gained popularity, a community developed around the language with a particular vocabulary. So older and more experienced programmers who were into object oriented early on may call things a bit different.

Parent and child is also used in describing other kinds of Is-A or Has-A relationships. For instance Parent window and Child window is also used for windowing systems in which a window, the Child, is contained within another window, the Parent. So the Parent window Has-A Child window.