In this paper we study the family of two-state Totalistic Freezing Cellular Automata (FTCA) defined over the triangular grids with von Neumann neighborhoods. We say that a Cellular Automaton is Freezing and Totalistic if the active cells remain unchanged, and the new value of an inactive cell depends only of the sum of its active neighbors. We study the family of FTCA in the context of asynchronous updating schemes (calling them FTACA), meaning that at each time-step only one cell is updated. The sequence of updated sites is called a sequential updating schemes. Given configuration, we say that a site is stable if it remains in the same state over any sequential updating scheme. In this context, we consider the Asynchronous Stability problem, consisting in decide whether there is a sequential updating scheme such that an inactive cell becomes active. We show that in this family the problem is NC, i.e. it can be solved by fast-parallel algorithms.