Q:

Does saving the super instance as a variable of the class is considered good practice?

Let's I have the class Dog which inherits from Animal, is it considered good practice to do the following:
class Dog(Animal):
    def __init__(self):
        self.__parentcls__ = super(Dog, self)

or should I just do super(Dog, self) every time I need to access my parent class?

A:

As the comments already indicate, there's no reason here to go against conventions.  Using super directly is already readable for python developers.
You should note that in python3, you use super(), not super(Dog, self), which naturally makes it more readable.
Also, one more disadvantage in your approach is that it creates a reference cycle (self references the super object, which references self, or: self.__parentcls__.__self__ is self).  While this isn't necessarilly a problem (python handles reference cycles well), it may impose some overhead (e.g. object will not get destroyed as soon as possible, only when gc cycles run).
EDIT: @Mehul aptly points out another real drawback of your approach: basically, it isn't subclass friendly.

