Some modifications of the Gehring lemma are required in the study of solutions to parabolic initial-boundary-value problems, The Gehring lemma assert that if a function satisfies the reverse Hölder-inequalities in a cube, then the integrability degree of this function in Q increases in this cube. Earlier, the author formulated some generalizations of the Gehring lemma and used them in the study of parabolic quasilinear systems with controlled nonlinearity orders. In this paper, the proof of these generalizations are given. On the basis of the modification of the Gehring lemma proposed by the author, the theorem on the reverse Hölder inequalities is formulated in a form convenient for obtaining L^p-estimates for the derivatives of solutions to parabolic problems. An application of this theorem is also demonstrated. Bibliography: 19 titles.