We introduce a deformed version of Dyck paths (DDP), where additional to the steps allowed for Dyck paths, ‘jumps’ orthogonal to the preferred direction of the path are permitted. We consider the generating function of DDP, weighted with respect to their half length, area and number of jumps. This represents the first example of a hierarchy of exactly solvable two-dimensional lattice vesicle models showing higher-order multi-critical points with scaling functions expressible via generalized Airy functions, as conjectured by John Cardy.