The leading contribution to the uncertainties of atmospheric neutrino flux calculations arise from the cosmic-ray nucleon flux and the production cross sections of secondary particles in hadron-air interactions. The data-driven model developed in this work parametrizes particle yields from fixed-target accelerator data. The propagation of errors from the accelerator data to the inclusive muon and neutrino flux predictions results in smaller uncertainties than in previous estimates, and the description of atmospheric flux data is good. The model is implemented as part of the MCEq package, and hence can be flexibly employed for theoretical flux error estimation at neutrino telescopes.