I run with multiple Python versions and thus multiple pip versions as well.
Everytime, however, you update pip, you’ll replace the standard pip command with the version you updated. So even pip3 install --upgrade pip will put a /usr/local/bin/pip in your system, messing up the Python 2 version.
Instead, I run pip as an (executable) module:
python3 -m pip search <package>
or
python2 -m pip search <package>
or even
python3.5 -m pip search <package>
This guarantees that your pip version always matches the Python version you want to use it for. It’s somewhat longer to type, but I prefer the expliciteness of it (which, I guess, follows the Zen of Python).
Note that updating pip:
python3.5 -m pip install --upgrade pip
will still install a Python 3.5 version in /usr/local/bin/pip, but I’m simply ignoring that. Just beware of (shell) scripts that execute pip directly.