No, paralysis can only be caused by damaging a nerve. Cartilage piercings, especially the ear(which is the most popular), do not run a risk of paralysis. Infection can be an issue however, but regular cleaning greatly reduces this risk.
It is not possible to get paralyzed if you pierce your cartilage because the nerves in your ear do not travel to your brain or any other major organ. The only risk in piercing your cartilage is heightened sensitivity or lack of sensitivity because of mussel damage. There is no visible damage if the mussel get infected.
I have 7 piercings, 3 of which I did myself (2 in cartilage, 1 in lobe), and I didn't have paralysis problems with any of them. People have told me you can become paralyzed from it, but I've never heard of it actually happening.
You can not get paralyzed by piercing your cartilage. The nerves in your ear don't travel to your brain or your spine, so there is no worry of become paralyzed after getting your ears pierced. There can be pain if one of these nerves is pierced, but nothing life threatening.