For recreation, I would like to count the number of times the path length, "area above," and area below the prime counting function, are simultaneously prime.
The following function is used to count the area below the prime counting function, where $p$ is prime:
$$I(p)= \int_{0}^{p} \pi(x) dx.$$
I will denote the triples as: $T=(A,P,Q),$
where $A=I(p),P=$path length, and $Q=$area above.
I found the first two triples:
$T_1=(11,11,17).$
$T_2=(37,19,41).$
If $p_n$ is the $n$th prime, your vertical axis is $n$, the fixed endpoints on your horizontal axis (i.e. where the path line rises) are $p_n$, so your total area is $n\cdot p_n$. You will only be able to divide the area into two prime values if $n$ is even. Your pathlength is $p_n+n$. You want $p_n+n$ to be prime. These relationships give you enough information to generate candidates. I haven't looked to see whether every candidate will be a solution.
Added by edit: Some values of $n$ such that $p_n+n$ is prime are: $4,6,18,22,24,26,32,34$ and so forth. You found the first two. What I can't promise without actually drawing the figures is whether the path will separate the area into sections whose areas are themselves prime.