I tend to carry at least a short length of rope pretty much all the time, even though rarely anything more substantial than paracord.

Paracord or equivalent is very handy all around, but I would never trust it as a climbing rope. It's simply not strong enough and leaves absolutely no room for error if you fall.

IMHO trying to substitute a proper climbing rope with paracord is inviting disaster. Sometimes, something might be worse than nothing. To be of any serious use, a climbing rope must be rated to hold much more than simply one's bodyweight. Maybe this link can explain it a little better:
http://www.alpineexposures.com/pages/faq-climbing-ropes-explained-test-uiaa