In this question we are counting labelled simple graphs. No concept of isomorphism is involved.
Let $G(n,e,t)$ be the number of labelled simple graphs with $n$ vertices, $e$ edges, and $t$ sets of three vertices which induce an odd number (1 or 3) of edges.
A block design enumeration problem can be solved if these numbers can be determined. However, I do not know any way to do that. I'm interested in any solution, like a summation, or generating function, or functional equation, or recurrence, or anything that allows computation of these numbers much faster than looking at all the graphs.
By considering the graph complement, we have $G(n,e,t) = G\left(n,\binom n2-e,\binom n3-t\right)$.
A generalisation would be to count labelled graphs with $n$ vertices, and $t_i$ sets of three vertices that induce $i$ edges for $i=0,1,2,3$, since the number of edges is easily determined from that information.
