Every inner edge of a triangle is subdivided into one or more 'short' segments by (the boundaries of) some other triangles on the opposite side. Each short segment is shared by exactly two triangles. Notice further that every short segment lies along a

unique segment of maximal length which is a concatenation of non-overlapping inner edges coming from the triangles on the same side of that segment. Hence, the total length of the short segments along one of maximal length is integer. Consequently, so is the total length $s$ of all short segments.
Clearly, every outer edge (lying on the boundary of $\Delta$) belongs to a single triangle, and the total length of all outer edges is the perimeter of $\Delta$.
Finally, let $t$ be the number of triangles, and let $S$ be the sum of their perimeters. Since the sides of each triangle all have an odd length, $t$ and $S$ have like parities. By the preceding, the perimeter of $\Delta$ is $S - 2s$, and the conclusion follows.