Print largest/smallest no (even, odd) from user defined list of given size
Your variables are long, without needing to be.
You don't have to define largest_even, amongst others, it'd be simpler to just return the tuple.
If you're targeting 3.4+, and I don't see why you wouldn't, you can use the default keyword. "New in version 3.4: The default keyword-only argument."


Leaving you simply with:

def largest_and_smallest(numbers):
    even = [n for n in numbers if (n % 2) == 0]
    odd  = [n for n in numbers if (n % 2) != 0]
    return (
        max(numbers, default=None),
        min(numbers, default=None),
        max(even, default=None),
        min(even, default=None),
        max(odd, default=None),
        min(odd, default=None)
    )


If you don't like the verbosity of max, you could use functools.partial to set default twice.

from functools import partial

def largest_and_smallest(numbers):
    min_ = partial(min, default=None)
    max_ = partial(max, default=None)
    even = [n for n in numbers if (n % 2) == 0]
    odd  = [n for n in numbers if (n % 2) != 0]
    return (
        max_(numbers),
        min_(numbers),
        max_(even),
        min_(even),
        max_(odd),
        min_(odd)
    )