It seems that at the moment Ubuntu 15.10 has a but with python3 and pip3.
As elaborated in this article.
The problem makes pip3 install to python3.5 while python3 is actually running python3.4 :(
Until a proper solution will be available via the updates you can do one of the following:
run
python3 -m pip install pymysql
instead of
pip3 install pymysql
(or any other package)
Now
import pymysql
should work in python3 and in idle3.
Alternatively, if you explicitly need 3.5 you can use explicit python3.5 instead of python3. but idle3 will still point to 3.4...