Even when the ILS serves opposite ends of the same runway there normally has to be an interlock that not only ensures only one transmitts at a time but if I remember correctly there is a specified delay between one being shut off and the other starting to transmitt.
Technically it would be possible to do the same with parallel runways however, one would have to ask why bother?
Such an arrangement removes the posibility of offering the "departure runway" to an arrival on an opportunity basic.
Such an arrangement also gives a single point of failure that has the potential to efffectively make both runways unavailable at the same moment i.e. runway A ILS fails and due to a switching failure runway B not working.
I don't think that ILS frequencies are at such a premium.
The only saving I could think of is in respect of the associated DME whereby a suitable located single DME could indicated zero at all four thresholds.
Overall while physically possible (with the appropriate protections), it does not sound like a good idea to me.