Random sequential adsorption (RSA) of particles of a particular shape is used in a large variety of contexts to model particle aggregation and jamming. A key feature of these models is the observed algebraic time dependence of the asymptotic jamming coverage as . However, the exact value of the exponent is not known apart from the simplest case of the RSA of monodisperse spheres adsorbed on a line (Renyi’s seminal “car parking problem”), where can be derived analytically. Empirical simulation studies have conjectured on a case-by-case basis that for general nonspherical particles, , where denotes the dimension of the domain, and the number of orientational degrees of freedom of a particle. Here, we solve this long-standing problem analytically for the case—the “Paris car parking problem.” We prove, in particular, that the scaling exponent depends on the particle shape, contrary to the original conjecture and, remarkably, falls into two universality classes: (i) for shapes with a smooth contact distance, e.g., ellipsoids, and (ii) for shapes with a singular contact distance, e.g., spherocylinders and polyhedra. The exact solution explains, in particular, why many empirically observed scalings fall in between these two limits.