The realization of triply degenerate points (TDPs) with exotic fermionic excitations has opened a new perspective for the understanding of our nature. Here we explore the coexistence of single unpaired TDP and multiple twofold Weyl points (WPs) and propose an experimental scheme with ultracold pseudospin-1 atomic gases trapped in optical lattices. We show that the predicted single unpaired TDP emerged by the interplay of quadratic spin-vector- and spin-tensor-momentum-coupling could possess a topological nontrivial middle band. This exotic TDP with mirror symmetry breaking is essential different from the recently observed TDPs that must appear in pairs due to the Nielsen-Ninomiya theorem and host the topological trivial middle band. Strikingly, the topologically protected Fermi-arc states directly connect the unpaired TDP with additional WPs, in contrast to the conventional Fermi-arc states that connect the same degeneracies of band degenerate points with opposite chirality. Furthermore, the different types of TDPs with unique linking structure of Fermi arcs can be readily distinguished by measuring spin texture along high-symmetry lines of the system. Our scheme provides a platform for emerging new fermions with exotic physical phenomena and versatile device applications.