We develop and analyze a parametric registration procedure for manifolds associated with the solutions to parametric partial differential equations in two-dimensional domains. Given the domain $Ω\subset \mathbb{R}^2$ and the manifold $M=\{ u_μ : μ\in P\}$ associated with the parameter domain $P \subset \mathbb{R}^P$ and the parametric field $μ\mapsto u_μ \in L^2(Ω)$, our approach takes as input a set of snapshots from $M$ and returns a parameter-dependent mapping $Φ: Ω\times P \to Ω$, which tracks coherent features (e.g., shocks, shear layers) of the solution field and ultimately simplifies the task of model reduction. We consider mappings of the form $Φ=\texttt{N}(\mathbf{a})$ where $\texttt{N}:\mathbb{R}^M \to {\rm Lip}(Ω; \mathbb{R}^2)$ is a suitable linear or nonlinear operator; then, we state the registration problem as an unconstrained optimization statement for the coefficients $\mathbf{a}$. We identify minimal requirements for the operator $\texttt{N}$ to ensure the satisfaction of the bijectivity constraint; we propose a class of compositional maps that satisfy the desired requirements and enable non-trivial deformations over curved boundaries of $Ω$; we develop a thorough analysis of the proposed ansatz for polytopal domains and we discuss the approximation properties for general curved domains. We perform numerical experiments for a parametric inviscid transonic compressible flow past a cascade of turbine blades to illustrate the many features of the method.