There's no such restriction. Perhaps you misread or misunderstood?
I keep reading that a smooth curve, say r(t) = (x(t),y(t)), is a curve such that r'(t) is continuous and r'(t) \neq 0.
I understand the "r'(t) is continuous" bit - so there are no "kinks" or anything in the curve (hence, smooth), but I don't understand why we require r'(t) \neq 0. Would anyone be able to explain this part to me please?
Oh I'm sorry, I see we're talking about a parametric curve. I'm not entirely sure about that, but my intuition says that if r'(t)=0 and we interpret t as time, then the curve "stops" at a particular point. I'm not sure why that is bad though.
I'll let someone else more familiar with parametric curve smoothness answer.