We develop a methodology for proving well-posedness in optimal regularity spaces for a wide class of nonlinear parabolic initial–boundary value systems, where the standard monotone operator theory fails. A motivational example of a problem accessible to our technique is the following system \partial _{t}u−\mathrm{div}\left(\nu (|\mathrm{∇}u|)\mathrm{∇}u\right) = −\mathrm{div}f with a given strictly positive bounded function \nu , such that \mathrm{\lim }_{k\rightarrow \infty }⁡\nu (k) = \nu _{\infty } and f \in L^{q} with q \in (1,\infty ) . The existence, uniqueness and regularity results for q \geq 2 are by now standard. However, even if a priori estimates are available, the existence in case q \in (1,2) was essentially missing. We overcome the related crucial difficulty, namely the lack of a standard duality pairing, by resorting to proper weighted spaces and consequently provide existence, uniqueness and optimal regularity in the entire range q \in (1,\infty ) . Furthermore, our paper includes several new results that may be of independent interest and serve as the starting point for further analysis of more complicated problems. They include a parabolic Lipschitz approximation method in weighted spaces with fine control of the time derivative and a theory for linear parabolic systems with right hand sides belonging to Muckenhoupt weighted L^{q} spaces.