Well that's odd... no wait, that's even!

CJam, 18 17 13 bytes

Thanks to aditsu for saving 4 bytes.

qW:O%eu~"eo"=

Try the test suite here. (The test suite is too long for the permalink. Just copy them from the challenge spec.)

Explanation

q     e# Read the input.
W:O   e# Push a -1 and store it in variable O.
%     e# Use the -1 to reverse the string, because CJam's stack-based nature and the
      e# commutativity of the operators means we can evaluate the code in postfix notation.
eu    e# Convert the string to upper case, turning 'e' into 'E' (a variable with even value
      e# 14) and 'o' into 'O' (which we've stored the odd value -1 in).
~     e# Evaluate the string as CJam code, leaving the result on the stack.
"eo"= e# Use the result as an index into the string "eo". CJam's indexing is cyclic so it
      e# automatically takes inputs modulo 2. Negative indices also work as expected.

Pyth, 16 14 bytes

@"eo".vjdXzGU9

Pyth can itself evaluate a string, that is in Pyth syntax. Therefore I replace e and o with 4 and 5. Then the evaluation will give me an even or odd number, and I can easily print the result.

Try it online: Demonstration or Test Suite

Explanation:

@"eo".vjdXzGU9   implicit: z = input string
         XzGU9   replace "e" in z with 4 and "o" with 5
       jd        put a space between each char
     .v          evaluate it (Pyth style)
@"eo"            and print "e" or "o"

Additional explanation to the replace. G is a variable initialized with the alphabet abc...xyz. U9 is the list [0, 1, ..., 8]. XzGU9 replaces the letters of the alphabet with the values of the list. So a gets replaced with 0, b with 1, ..., e with 4, ..., i with 8, j with 0, ..., and o with 5. Therefore I e gets replaced with an even number and o with an odd number. All the other replacements have no effect at all.

Perl, 50 45 40 characters

(39 characters code + 1 character command line option.)

1while s/\+oe|\+eo|\*oo/o/||s/\W\w\w/e/

Sample run:

bash-4.3$ echo -n '**o++*ee*++eoe*eo+eoo' | perl -pe '1while s/\+oe|\+eo|\*oo/o/||s/\W\w\w/e/'
o