An edge partitioning problem on cubic graphs
Here is a suggestion for how to show it's NP hard. I don't know if this works or not. First, consider the same problem on multigraphs. NP hardness may be easier to prove there. Try reducing from cubic MAX CUT which is NP hard even to approximate (Berman and Karpinski "On Some Tighter Inapproximability Results"). Divide each edge into two and at each of the new degree-2 vertices attach a vertex with self-loop. Does your maximum partition correspond to a maximum cut?

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Here is a bit more explanation.

(1)
The problem of maximizing (number of sources+number of sinks) over all orientations of a cubic graph is related to MAXCUT by some linear function. This requires showing some corersspondence between maximal cuts and sources-and-sinks-maximal orientations. In one direction, in a maximum cut (U,V), we can orient all edge from U to V. The internal edges E(U) form a matching, so these can be oriented arbitrarily and similarly for E(V), and the total number of sources and sinks is some linear function of the size of the cut. In the other direction, given a sources-and-sinks-maximal orientation, the partition U=vertices of in-degree 0 or 1, V=vertices of in-degree 2 or 3 gives a cut.

(2)
In the edge-bisecting transformation I described above, in an optimal configuration each loop is coloured the same as the edge next to it, and w.l.o.g that edge is coloured the same as the some other (non-loop) edge next to that. So each bisected edge has one colour coming from its attached loop and one other colour. This corresponds to an orientation and (1) applies.