  This work discusses parabolic Muckenhoupt weights on spaces of homogeneous
type, i.e.\ quasi-metric spaces with both a doubling measure and an additional
monotone geodesic property. The main results include a characterization in
terms of weighted norm inequalities for parabolic maximal operators, a reverse
H\"older inequality, and a Jones-type factorization result for this class of
weights. The connection between the space of parabolic bounded mean oscillation
and parabolic Muckenhoupt weights is studied by applying a parabolic
John--Nirenberg lemma. A Coifman--Rochberg-type characterization of the space
of parabolic bounded mean oscillation in terms of parabolic maximal functions
is also given. The main challenges in the parabolic theory are related to the
time lag in the estimates. The results are motivated by the corresponding
Euclidean theory and the regularity theory for parabolic variational problems
on metric measure spaces.
