find_an() looks like an internal function, used by arithmetic_partial_sum(). If it is not for external use, it should be named with a leading underscore, to suggest it is private.

arithmetic_partial_sum() appears to handle only integer values, yet it return a floating-point value (1275.0 in the built-in example). It should return an integer, since it is adding up integers. Use the Python3.x integer-division operator: //.

    n = 1 + (an - a1) // d
    S = n * (a1 + an) // 2

Or, not assume the terms are integer, and use float(...) instead.

geometric_partial_sum() fails if the first 2 values are not int values:

>>> geometric_partial_sum(f"1+{1/3}+{1/9}+{1/27}+{1/81}")
Traceback (most recent call last):
  File "<pyshell#1>", line 1, in <module>
    geometric_partial_sum(f"1+{1/3}+{1/9}+{1/27}+{1/81}")
  File "...\partial_sum.py", line 63, in geometric_partial_sum
    r = int(series[1]) / a1
ValueError: invalid literal for int() with base 10: '0.3333333333333333'

You should convert the terms to floating-point values, not integers:

    a1 = float(series[0])
    r = float(series[1]) / a1

find_an() assumes \$a_n\$ is immediately after the '.' term, so will fail with:

arithmetic_partial_sum("3+7+11+15+.+97+99")
arithmetic_partial_sum("3+7+11+15+17")

Why not just retrieve the last term?

def find_an(parsed_series):
    return int(parsed_series[-1])

Now the following all succeed and return the correct values

arithmetic_partial_sum("3+7+11+15+.+97+99")
arithmetic_partial_sum("3+7+11+15+...+97+99")
arithmetic_partial_sum("3+7+11+15+17")

The """docstrings""" for arithmetic_partial_sum() and geometric_partial_sum() appear unhelpful. For example:

>>> help(arithmetic_partial_sum)
Help on function arithmetic_partial_sum in module __main__:

arithmetic_partial_sum(series)
    Returns the partial sum of an arithmetic series
    
    Formula:
        S = n( (a1 + an) / 2 )
    
    Find an:
        an = a1 + (n - 1)d
    
    Find n:
        n = 1 + ( (an - a1) / d )
    
    :param series: Arithmetic series to solve

The function is not returning an or n. Even the formula is not particularly helpful. """docstrings""" should tell a user how to use the function. For example (adding Python 3.6 type hints as well):

def arithmetic_partial_sum(series:str) -> int:
    """
    Returns the sum of an arithmetic series

    Example:
        s = arithmetic_partial_sum("1+3+5+.+99")  # returns 2500
        
    :param series: A string representing the arithmetic series to solve
    """

Now type help(arithmetic_partial_sum):

>>> help(arithmetic_partial_sum)
Help on function arithmetic_partial_sum in module __main__:

arithmetic_partial_sum(series: str) -> int
    Returns the sum of an arithmetic series
    
    Example:
        s = arithmetic_partial_sum("1+3+5+.+99")  # returns 2500
        
    :param series: A string representing the arithmetic series to solve

The user is told the function takes a string and returns an integer. The format of the string should be clear from the example.