Modeling of laser-induced photoionization is a key component in simulations of ultrafast laser ablation of transparent solids. Available analytical and phenomenological models consider the ionization by promoting valence band electrons to a conduction band via simultaneous absorption of several laser photons of the same energy defined at central wavelength of laser-pulse spectrum. That assumption corresponds to nearly zero spectrum bandwidth and is not true for ultrashort few-cycle pulses. Treatment of the few-cycle-pulse photoionization by numerical methods, e.g., time-dependent density-functional theory, meets substantial difficulties when modeling of ionization-rate scaling with multiple laser-pulse and material parameters. To address those gaps, we report an analytical approach to evaluation of the photoionization rate by a few-cycle laser pulse with nonnegligible spectral bandwidth. We assume that the pulse spectrum supports the few-cycle pulse duration, but is narrow enough to prevent quantum interference between multiphoton transitions of different orders. We outline the calculation procedure, report examples of simulations with the proposed model, and discuss some novel physics of the photoionization. Our model includes dependence of the photoionization rate on carrier-envelope phase and delivers significantly higher rates compared with the Keldysh model. We interpret those results in terms of laser-driven variations of effective band gap, opening of extra channels of the multiphoton absorption, and involvement of a continuous range of electron states determined by pulse spectrum width. Based on the reported model, we discuss new options to control threshold of ultrafast laser ablation via carrier-envelope and pulse-shape scaling of the photoionization rate.