We obtain the boundedness of parabolic fractional integral operators TΩ,α\documentclass[12pt]{minimal}
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\begin{document}$$T_{\Omega ,\alpha }$$\end{document} with variable kernels Ω(·,·)\documentclass[12pt]{minimal}
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\begin{document}$$\Omega (\cdot ,\cdot )$$\end{document} belonging to L∞(Rn)×Ls(Sn-1),s>n/(n-α)\documentclass[12pt]{minimal}
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\begin{document}$$L^{\infty }({\mathbb {R}^n}) \times L^{s}({\mathbb {S}}^{n-1}), s>n/(n-\alpha )$$\end{document}, and their commutators [b,TΩ,α]\documentclass[12pt]{minimal}
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\begin{document}$$[b,T_{\Omega ,\alpha }]$$\end{document} with BMO functions in variable exponent generalized Morrey spaces Mp(·),φ\documentclass[12pt]{minimal}
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\begin{document}$$M^{p(\cdot ),\varphi }$$\end{document} and variable exponent vanishing generalized Morrey spaces VMp(·),φ\documentclass[12pt]{minimal}
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\begin{document}$$\textrm{VM}^{p(\cdot ),\varphi }$$\end{document}. We find the sufficient conditions on the pair (φ,ψ)\documentclass[12pt]{minimal}
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\begin{document}$$(\varphi ,\psi )$$\end{document} which ensures the boundedness of the operators TΩ,α\documentclass[12pt]{minimal}
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\begin{document}$$T_{\Omega ,\alpha }$$\end{document} and [b,TΩ,α]\documentclass[12pt]{minimal}
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\begin{document}$$[b,T_{\Omega ,\alpha }]$$\end{document} from Mp(·),φ\documentclass[12pt]{minimal}
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\begin{document}$$M^{p(\cdot ),\varphi }$$\end{document} to Mq(·),ψ\documentclass[12pt]{minimal}
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\begin{document}$$M^{q(\cdot ),\psi }$$\end{document} and from VMp(·),φ\documentclass[12pt]{minimal}
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\begin{document}$$\textrm{VM}^{p(\cdot ),\varphi }$$\end{document} to VMq(·),ψ\documentclass[12pt]{minimal}
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\begin{document}$$\textrm{VM}^{q(\cdot ),\psi }$$\end{document}.
