By its very nature, this question cannot expect a definitive answer but here are some suggestions.

1.  For a curve with parametrisation of the form $(F(t),f(t))$ with $F$ a primitive of $f$ it is the case if $f$ is injective or, better, if and only if $F(s)\neq F(t)$ whenever $f(s)=f(t)$.

2.  Many important curves have parametrisations of this form (circle, cycloid, catenary, ....).

3.  In a certain sense "every" curve has such a parametrisation.  More precisely, consider the curve with parametrisation $(x(s),y(s))$.  Since self-intersection is preserved under diffeomorphisns, we can suppose that the curve lies in the upper half plane.  Under the new parameter $t$ where the latter is, as a function of $s$, the primitive of $\frac{x'(s)}{y(s)}$, the parametrisation will have the above form.

4.  Of course, this will not work universally since this $t$ will not always be a 
reparametrisation but it will be in many concrete situations, e.g., if $x$ is strictly monotone.