In issues like this, there is more than just punctuation that is the problem.
I would like to thank my parents, Nelson Mandela and Mother Theresa."
Here, on the other hand, its presence is:
"I would like to thank my father, the Pope, and Dame Judi Dench."
In cases of the title of parent or father there could follow a further clarification of who those are. This could lead to expectation.
Then there is the issue of clarity which is closely related.
In the first instance.
I would like to thank my parents and Nelson Mandela and Mother Theresa.
Reduces the expectations beyond parents and removes all commas and I think is much clearer.
Granting that you wouldn't expect someones father to be the Pope--there are a lot of other titles that might fit such as president.
So, once again.
I would like to thank my father and the Pope and Dame Judi Dench.
While the examples are humorous, I think they beg to be mended this way for clarity and anything else is open for misinterpretation.
To preserve the comma you might best go this way.
I would like to thank the Pope, Dame Judi Dench and my father.
Though it might be politically correct to start with my father.
The same for the line with parents.
I would like to thank Nelson Mandela, Mother Theresa and my parents.
What's of interest is...
If you use something like sister or brother you can get it done with...
I would like to thank my brother John.
I would like to thank my sister Karen.
However it becomes suspect when you use...
I would like to thank my brother John, Karen and Lisa.
This seems acceptable grammatically because it is singular brother, however for clarity you might still want to use.
I would like to thank my brother John and Karen and Lisa.
If the others are sisters you might need...
I would like to thank my brother John and sisters, Karen and Lisa.
Oddly enough grammatically it might seem acceptable to use...
I would like to thank my parent Dave. (this seems to be because you might want to specify which of the two parents and could be necessary).
However not.
I would like to thank my father Dave.
It wants to be.
I would like to thank my father, Dave. (I think because you should only have one father so Dave is not as necessary).