ABSTRACT A class of long gamma-ray bursts (GRBs) presenting light curves with an extended plateau phase in their X-ray afterglows obeys a correlation between the rest-frame end-time of the plateau, Ta, and its corresponding X-ray luminosity, La, (Dainotti et al). In this work we perform an analysis of a total sample of 176 Swift GRBs with known redshifts, exhibiting afterglow plateaus. By adding a third parameter that is the peak luminosity in the prompt emission, Lpeak, we discover the existence of a new three-parameter correlation. The scatter of data about this plane becomes smaller when a class-specific GRB sample is defined. This sample of 122 GRBs is selected from the total sample by excluding GRBs with associated supernovae (SNe), X-ray flashes and short GRBs with extended emission. With this sample the three-parameter correlation identifies a GRB "fundamental plane." Moreover, we further limit our analysis to GRBs with light curves with good data coverage and almost flat plateaus, 40 GRBs forming our "gold sample." The intrinsic scatter, , for the three-parameter correlation for this last sub-class is more than two times smaller than the value for the one, making this the tightest three-parameter correlation that involves the afterglow plateau phase. Finally, we also show that a slightly less tight correlation is present between Lpeak and a proxy for the total energy emitted during the plateau phase, , confirming the existence of an energy scaling between the prompt and afterglow phases.