Equal sums

Let $P =$ sum of pink numbers, $Y =$ sum of yellow numbers, and $S =$ sum of the numbers across a line. When we add up all of the numbers on each line, we count each pink number twice and each yellow number once. Thus,

$2P + Y = 4S.$

The sum $P + Y$ is equal to the sum of all the numbers, which is $1 + 2 + \dots + 8 = 36.$ So, we can simplify this equation into

$P + 36 = 4S.$

As the sum of four distinct integers between 1 and 8 inclusive, $P$ can take on any value between 10 and 26 inclusive. We note that $P$ must be a multiple of 4, since $S$ must be an integer. Also, in order to maximize $Y,$ we must let $P$ be as small as possible. The smallest multiple of 4 between 10 and 26 is 12, so the maximum value of $Y = 36 - P = 36 - 12 = \boxed{24}.$

Here's a configuration with $Y = 24$ :

$\begin{array}{ccc} \boxed{1} & \boxed{5} & \boxed{6} \\ \boxed{8} & & \boxed{4} \\ \boxed{3} & \boxed{7} & \boxed{2} \end{array}$