We have studied collisional perturbations of radio-frequency (rf) electric-dipole (E1) transitions between the nearly degenerate opposite-parity levels in atomic dysprosium (Dy) in the presence of 10 to 80 $μ$Torr of H$_\text{2}$, N$_\text{2}$, He, Ar, Ne, Kr, and Xe. Collisional broadening and shift of the resonance, as well as the attenuation of the signal amplitude are observed to be proportional to the foreign-gas density with the exception of H$_2$ and Ne, for which no shifts were observed. Corresponding rates and cross sections are presented. In addition, rates and cross sections for O$_2$ are extracted from measurements using air as foreign gas. The primary motivation for this study is the need for accurate determination of the shift rates, which are needed in a laboratory search for the temporal variation of the fine-structure constant [A. T. Nguyen, D. Budker, S. K. Lamoreaux, and J. R. Torgerson, Phys. Rev. A \textbf{69}, 22105 (2004)].