  Newton-Sobolev spaces, as presented by N. Shanmugalingam, describe a way to
extend Sobolev spaces to the metric setting via upper gradients, for metric
spaces with `sufficient' paths of finite length. Sometimes, as is the case of
parabolic metrics, most curves are non-rectifiable. As a course of action to
overcome this problem, we generalize some of these results to spaces where
paths are not necessarily measured by arc length. In particular, we prove the
Banach character of the space and the absolute continuity of these Sobolev
functions over curves. Under the assumption of a Poincar\'e-type inequality and
an arc-chord property here defined, we obtain the density of some Lipschitz
classes, relate Newton-Sobolev spaces to those defined by Hajlasz by means of
Hajlasz gradients, and we also get some Sobolev embedding theorems. Finally, we
illustrate some non-standard settings where these conditions hold, specifically
by adding a weight to arc-length and specifying some conditions over it.
