Took a look at the picture. It appears the answer is no. If the sum of the numbers in adjacent triangles is odd, then one triangle contains an odd number while the other contains an even. Light green triangles are adjacent only to dark green triangles and vice-versa. So if light green triangles contain odd numbers, the dark green triangles contain even numbers. There are equal numbers of odd and even numbers in the range 1 through 16, but there are different numbers of dark and light green triangles.
By the pigeonhole principle, one of the dark green triangles must contain a number of the wrong parity. Therefore, there will be an even sum somewhere.