The method for the synthesis of ethyl 4-[3-(ethoxycarbonyl)-1-phenyl-1H-pyrazol-4-yl]-2-oxo/thioxo-1,2,3,4-tetrahydropyrimidine-5-carboxylates was developed, and the bioactivity of the obtained compounds against infectious agents was predicted. Docking studies of the synthesised compounds were performed on pteridine reductase 1 (PTR1) of Leishmania, dihydrofolate reductase–thymidylate synthase of Trypanosoma cruzi, and human dihydrofolate reductase. The results demonstrated that the investigated compounds exhibit high affinity for these enzymes. Moderate selectivity relative to human DHFR was also observed. In addition, the predicted drug-likeness, ADME-Tox parameters, and toxicity profiles suggest the potential of the synthesised compounds for further pharmacological development.