I'm happy with the current wording.  Attempts to over-explain will make
it harder to understand.  The term "parent" is used loosely to mean one
of the preceding classes in the inheritance diagram except for the class
of the caller.  To make that interpretation clearer, the next sentence
specifically references the __mro__ attribute.  Further explanations of
search-order and multiple inheritance should be done somewhere else.   
The docs for super() are not are the right place to cover all aspects of
multiple inheritance.

In the case of single inheritance, the term "parent" is exactly correct.
 In the case of multiple inheritance, the notion of "parent" becomes
that-which-precedes-self-in-the-dynamically-computed-search-order.  IMO,
the loose use of "parent" is appropriate in a first sentence. 
Otherwise, the whole doc entry becomes incomprehensible from the start.  

If someone adds a fuller explanation of multiple-inheritance to the
docs, I would be happy to link to it but I don't want to make the docs
for super() any more inpenatrable that they already are.