Yes, it is possible, but the mechanics of such arrangements invariably require your parent unit to keep paying for your training with your host unit. Something which, understandably, not all units are happy to do.
You'll need to ensure that your host unit is happy to have you tag along and that your parent unit is willing to pay you for training days and parade nights there. PSAOs are more relevant than PSIs in this, as they will paysheet information transfer, but both OCs will need to sign off on it first.
In summary, if everyone's happy to play ball, it shouldn't be huge drama.
I'm in a dilemma of whether to stay where I am in a non-deployable role(according to RLC HQ) in a role that now, just after finishing class 3, my unit tell me they have no use for, or move to a unit where I may be relevant( or as relevant as a STAB can be) at least stand a chance of deploying.