This work presents a symmetry-based analytical study of a frictional Aw-Rascle-Zhang (ARZ) traffic-flow model with Chaplygin-type pressure laws. The governing system is formulated as a nonlinear hyperbolic balance law with a Coulomb-like friction term. Local Lie point symmetries and invariant reductions are considered for the Classical, Generalized, and Modified Chaplygin pressure laws. For the classical Chaplygin case, the analysis is further extended through conservation-law potentials to identify nonlocal symmetry structures and associated invariant solutions. The reductions yield explicit solution families, including inverse-time density profiles, nonlinear-power and logarithmic velocity corrections, branches with moving singular density boundaries, and bounded traveling fronts in the frictionless limit. The regularity and physical admissibility of these solutions are assessed on explicitly stated domains through density positivity and finite velocity behavior. The obtained profiles provide exact analytical benchmarks for comparison and numerical validation. Stability, entropy admissibility, shock formation, and evolution from general initial data remain outside the scope of the present symmetry-based analysis.