`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+.+95+99")
    arithmetic_partial_sum("3+7+11+15+19")

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+.+95+99")
    arithmetic_partial_sum("3+7+11+15+...+95+99")
    arithmetic_partial_sum("3+7+11+15+19")

-------

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.