Rita, I may be misunderstanding something
EDIT: here is a short, but wouldn't non-elementary argument to prove that the following work?threefold in question is uniruled:
The canonical class of your complete intersection is $(-6+2+3)H=-H$ where $H$ is the hyperplane section. Therefore this is a Fano and hence unirational uniruled by the Miyaoka-Mori criterion, which is essentially an application of Bend and Break.
Did you want something more elementary? One can also get this by running the mmp on it, but ultimately it boils down to the same thing. Of course, most likely Enriques's proof is more specific to the situation.