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In this paper, we prove the Lorentz space $L^{q,p}$-estimates for gradients of very weak solutions to the linear parabolic equations with $\mathbf{A}_q$-weights $$u_t-\operatorname{div}(A(x,t)\nabla u)=\operatorname{div}(F),$$ in a bounded domain $\Omega\times (0,T)\subset\mathbb{R}^{N+1}$, where $A$ has a small mean oscillation, and $\Omega$ is a Lipchistz domain with a small Lipschitz constant.

Gradient weighted norm inequalities for very weak solutions of linear parabolic equations with BMO coefficients