A relation is an equivalence relation if and only if it satisfies three properties:
(1) symmetry ($h_1Rh_1$ for any $h_1$),
(2) reflexivity ($h_1Rh_2$ implies $h_2Rh_1$), and
(3) transitivity ($h_1Rh_2$ and $h_2Rh_3$ imply $h_1Rh_3$)
Both $h1$ and $h2$ are clearly reflexive (you have parents in common with yourself) and symmetric (if I have a parent in common with you, then you have a parent in common with me), but as you noted $h1$ is not transitive: we can imagine a situation where $h1$ and $h2$ have the same mother but not the same father, $h2$ and $h3$ have the same father but not the same mother. $R_2$, on the other hand, is transitive, so $R_2$ is an equivalence relation.
It should be noted that $R_1$ not being transitive (and more in general, relation properties) depends on the set you are considering. It's safe to say that the scenario that served as a counterexample happened somewhere on this planet, but on a smaller set $R_1$ could be transitive (for example, it might be so among the people you know personally).
The wikipedia article on the topic is well written, and has more examples and explanations about equivalency relations.