Oxygen interstitial solutes have traditionally been thought to occupy only octahedral interstices in body-centered cubic (bcc) metals. However, there is ongoing debate about the competition between tetrahedral and octahedral interstitial sites for oxygen solutes in multi-principal element alloys, especially following recent experimental evidence obtained using deep-sub-angstrom-resolution electron ptychography. The driving force and atomistic mechanism behind the observed preference switch remain unclear. In this study, we systematically investigate the competition between tetrahedral and octahedral sites for oxygen, nitrogen, and carbon solute atoms in bcc NbTiZr alloys using density functional theory calculations. At dilute concentrations, these small interstitial solutes are observed to preferentially reside in octahedral interstices. Notably, we demonstrate that solutes can switch from octahedral to tetrahedral sites with increasing lattice expansion, interstitial solute concentration, and spatial heterogeneity. This transition is explained by a crossover in elastic strain energy associated with solutes in the two competing interstices. Our findings provide a deeper understanding of the behavior of small interstitial solutes and their solid solution strengthening effects in bcc multi-principal element alloys.