This work presents a comprehensive framework for capturing bifurcating phenomena and detecting bifurcation curves in nonlinear multi-parametric partial differential equations, where the system exhibits multiple coexisting solutions for given values of the parameters. Traditional continuation methods, employing the previously computed solution as the initial guess for the next parameter value, are usually very inefficient, since small step sizes increase computational cost, while larger steps could jeopardize Newton's method convergence jumping to a different solution branch or missing the bifurcation point. To address these challenges, we propose a framework that combines arclength continuation, adaptively selecting parameter values in a multi-parametric scenario, with the deflation technique, discovering multiple branches, to construct complete bifurcation diagrams. In particular, the arclength continuation method is designed to handle multi-parametric scenarios, where the parameter vector $\lambda \in \mathbb{R}^p$ traces a curve $g(\lambda)$ within a $p$-dimensional parameter space. The deflation technique is very crucial to compute coexisting solution branches, by modifying the PDE residual to penalize convergence to previously identified solutions, it guides the solver towards discovering new ones even with the same initial guess. Moreover, detecting bifurcation curves and surfaces, respectively for $p=2$ and $p=3$, is critical for understanding the qualitative behavior of such systems for multiple parametric configurations. We introduce a zigzag path-following strategy to robustly track the bifurcation curves, demonstrating its performance on two benchmark problems: the Bratu equation and the Allen–Cahn equation, in both single- and multi-parametric settings.