We study the ground-state density profile of a spin-orbit-coupled f = 2 spinor condensate in a quasi-one-dimensional trap. The Hamiltonian of the system is invariant under time reversal but not under parity. We identify different parity- and time-reversal symmetry-breaking states. The time-reversal symmetry breaking is possible for degenerate states. A phase separation among densities of different components is possible in the domain of time-reversal symmetry breaking. Different types of parity- and time-reversal symmetry-breaking states are predicted analytically and studied numerically. We employ numerical and approximate analytic solutions of a mean-field model in this investigation to illustrate our findings.Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)