A host-parasite model is considered for a population of cells that can be of two types, A or B, and exhibits unilateral reproduction: while a B-cell always splits into two cells of the same type, the two daughter cells of an A-cell can be of any type. The random mechanism that describes how parasites within a cell multiply and are then shared into the daughter cells is allowed to depend on the hosting mother cell as well as its daughter cells. Focusing on the subpopulation of A-cells and its parasites, the model differs from the single-type model recently studied by Bansaye (2008) in that the sharing mechanism may be biased towards one of the two types. Main results are concerned with the nonextinctive case and provide information on the behavior, as $n\to\infty$, of the number A-parasites in generation n and the relative proportion of A- and B-cells in this generation which host a given number of parasites. As in (Bansaye,2008), proofs will make use of a so-called random cell line which, when conditioned to be of type A, behaves like a branching process in random environment.