In this article we study the bivariate truncated moment problem (TMP) of degree 2k on the union of parallel lines. First we present an alternative proof of Fialkow's solution [28] to the TMP on the union of two parallel lines (TMP–2pl) using the solution of the truncated Hamburger moment problem (THMP). We add a new equivalent solvability condition, which is then used together with the THMP, to solve the TMP on the union of three parallel lines (TMP–3pl), our second main result of the article. Finally, we establish a sufficient condition for the existence of a solution to the TMP on the union of n parallel lines in the pure case, i.e. when the moment matrix Mk is of the highest possible rank, or equivalently the only column relations come from the union of n lines. The condition is based on the feasibility of a certain linear matrix inequality, corresponding to the extension of Mk by adding rows and columns indexed by some monomials of degree k+1. The proof is by induction on n, where n≥2 and for the base of induction n=2 we use the solution of the TMP–2pl.