In the first part of this paper, we determine the asymptotic subgroup growth of the fundamental group of a torus knot complement. In the second part, we use this to study random finite degree covers of torus knot complements. We determine their Benjamini-Schramm limit and the linear growth rate of the Betti numbers of these covers. All these results generalise to a larger class of lattices in $\mathrm{PSL}(2,\mathbb{R})\times \mathbb{R}$. As a by-product of our proofs, we obtain analogous limit theorems for high degree random covers of non-uniform Fuchsian lattices with torsion.