  Praseodymium (Pr$^{3+}$) ions doped in the site 1 of yttrium orthosilicate
(Y$_2$SiO$_5$) has been widely employed as the photonic quantum memory due to
their excellent optical coherence and spin coherence. While praseodymium ions
occupying the site 2 in Y$_2$SiO$_5$ crystal have better optical coherence as
compared with those at site 1, which may enable better performance in quantum
memory. Here we experimentally characterize the hyperfine interactions of the
ground state $^3$H$_4$ and excited state $^1$D$_2$ of Pr$^{3+}$ at site 2 in
Y$_2$SiO$_5$ using Raman heterodyne detected nuclear magnetic resonance. The
Hamiltonians for the hyperfine interaction are reconstructed for both ground
state $^3$H$_4$ and excited state $^1$D$_2$ based on the Raman heterodyne
spectra in 201 magnetic fields. The two-pulse spin-echo coherence lifetime for
the ground state is measured to be 2.6$\pm$0.1 ms at site 2 with zero magnetic
field, which is more than five times longer than that at site 1. The magnetic
fields with zero first order Zeeman shift in the hyperfine transition for
Pr$^{3+}$ at site 2 in Y$_2$SiO$_5$ are identified.
