In this paper, we obtain the exact values of several domination parameters for the direct product of a complete graph with a path or a cycle. Specifically, we determine the domination number, independent domination number, $[1,2]$-domination number, secure domination number, and 2-domination number for this family of graphs. We show that, in these graphs, the independent domination number and the $[1,2]$-domination number coincide with the domination number, while the secure domination number coincides with the 2-domination number. Additionally, as a consequence of our findings, we provide counterexamples to disprove some erroneous results in the literature.