Flow polytopes are fundamental objects in algebraic combinatorics. In this paper, we study the enumeration of lattice points in flow polytopes associated with $(a_1,a_2)$-caracol graphs on $n+2$ vertices. Our main result establishes a closed-form expression for the number of lattice points by constructing an explicit combinatorial bijection between Dyck paths and the integer lattice points of the two-parameter family of polytopes $(a_1,a_2)$-Car ${}_{n+1}$, using pseudo-ladder diagrams together with vector partition techniques. When $a_2=1$, the lattice point sequence of caracol polytopes coincides with the OEIS sequence A126216 (The On-Line Encyclopedia of Integer Sequences), which enumerates Schröder paths of semilength $n$ with exactly $k$ peaks. Furthermore, we establish a bijection between Schröder paths and the integer lattice points of the two-parameter family of polytopes $(a_1,a_2)$-Car ${}_{n+1}$. All bijections are implemented as explicit algorithms in Python, with the complete source code provided in the appendix to ensure reproducibility.