  In this paper we explicitly describe the symbolic powers of curves ${\mathcal
C}(q,m)$ in ${\mathbb P}^3$ parametrized by
  $( x^{d+2m}, x^{d+m} y^m, x^{d} y^{2m}, y^{d+2m})$, where $q,m$ are positive
integers, $d=2q+1$ and $\gcd(d,m)=1$. The defining ideal of these curves is a
set-theoretic complete intersection. We show that the symbolic blowup algebra
is Noetherian and Gorenstein. An explicit formula for the resurgence and the
Waldschmidt constant of the prime ideal ${\mathfrak p}:={\mathfrak p}_{ {
\mathcal C}(q,m) }$ defining the curve ${\mathcal C}(q,m)$ is computed. We also
give a formula for the Castelnuovo-Mumford regularity of the symbolic powers
${\mathfrak p}^{(n)}$ for all $n \geq 1$.
