There's more ways than one to solve this, just as there's more ways than one to skin a cat.
Two paralle lines have the same slope, so the desired new line looks exactly like the given line 3x + 5y = 6, except that the constant, 6, is replaced by another value.
Write out 3x + 5y = c
Replace x with 0 and y with 3. Then 3(0) + 5(3) = c, and c = 15.
Thus, the equation of the new, parallel, line is 3x + 5y = 15.