Aneuploidy is generally considered harmful, but in some microorganisms, it can act as an adaptive mechanism against environmental stress. Here, we use Leishmania-a protozoan parasite with remarkable genome plasticity-to study the early steps of aneuploidy evolution under high drug pressure (using antimony or miltefosine as stressors). By combining single-cell genomics, lineage tracing with cellular barcodes, and longitudinal genome characterization, we reveal that aneuploidy changes under antimony pressure result from polyclonal selection of pre-existing karyotypes, complemented by further and rapid de novo alterations in chromosome copy number along evolution. In the case of miltefosine, early parasite adaptation is associated with independent point mutations in a miltefosine transporter gene, while aneuploidy changes only emerge later, upon exposure to increased drug levels. Therefore, polyclonality and genome plasticity are hallmarks of parasite adaptation, but the scenario of aneuploidy dynamics depends on the nature and strength of the environmental stress as well as on the existence of other pre-adaptive mechanisms.