  This paper extends and complements the existing theory for the parabolic
Muckenhoupt weights motivated by one-sided maximal functions and a doubly
nonlinear parabolic partial differential equation of $p$-Laplace type. The main
results include characterizations for the limiting parabolic $A_\infty$ and
$A_1$ classes by applying an uncentered parabolic maximal function with a time
lag. Several parabolic Calder\'on-Zygmund decompositions, covering and chaining
arguments appear in the proofs.
