X-ray circular magnetic dichroism (XMCD), longitudinal ($chi_{ac}$) and transverse (TS) ac magnetic susceptibility have been measured in several members of the $R$Co$_2$ series ($R$ = Dy, Ho, and Tm) as a function of temperature and applied magnetic field. We show that parimagnetism is a general behavior along the $R$Co$_2$ ferrimagnetic series ($R$ being a heavy rare earth ion). XMCD results evidence the presence of two compensation temperatures, defining two different parimagnetic configurations, which is a fully unexpected result. The inverse $chi_{ac}$ curve exhibits a deviation from Curie-Weiss behavior which is recovered under applied magnetic field. The large excess of polarizability above the critical temperature proves the existence of an enhanced effective moment due to the presence of short range magnetic correlations, which are also observed in TS measurements. The combination of TS and XMCD measurements allows to depict new magnetic phase diagrams for the $R$Co$_2$ series. A new scenario allowing to understand the observed phenomenology as a Griffiths phase-like behavior is proposed, where the amorphous $R$Co$_2$ represents the undiluted system case.
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