  We consider quasilinear parabolic equations with measurable coefficients when
the right-hand side is a signed Radon measure with finite total mass, having
$p$-Laplace type: $$u_t - \textrm{div} \, \mathbf{a}(Du,x,t) = \mu \quad
\textrm{in} \ \Omega \times (0,T) \subset \mathbb{R}^n \times \mathbb{R}.$$ In
the singular range $\frac{2n}{n+1} <p \le 2-\frac{1}{n+1}$, we establish
regularity estimates for the spatial gradient of solutions in the Marcinkiewicz
spaces, under a suitable density condition of the right-hand side measure.
