We perform laser spectroscopy of dysprosium monoxide (DyO) to determine the hyperfine structure of the ground X8 and excited [17.1]7 states in the ¹⁶¹Dy and ¹⁶³Dy isotopologues. These dysprosium nuclei have nonzero nuclear spin and dynamical octupole deformation, providing them high sensitivity to time-reversal-violating new physics via the nuclear Schiff moment (NSM). The DyO molecule was recently identified as being amenable to optical cycling─the basis for many laser cooling and quantum control techniques─which makes it a practical candidate for NSM searches. The measurements reported here are prerequisites to implementing optical cycling, designing precision measurement protocols, and benchmarking calculations of molecular sensitivity to symmetry-violating effects. The measured hyperfine parameters are interpreted using simple molecular orbital diagrams and show excellent agreement with relativistic quantum chemical calculations.