  We investigate parity-time ($\mathcal{PT}$) phase transitions in open quantum
systems and discuss a criterion of Liouvillian $\mathcal{PT}$ symmetry proposed
recently by Huber \textit{et al}. [J. Huber \textit{et al}., SciPost Phys.
$\textbf{9}$, 52 (2020)]. Using the third quantization, which is a general
method to solve the Lindblad equation for open quadratic systems, we show, with
a proposed criterion of $\mathcal{PT}$ symmetry, that the eigenvalue structure
of the Liouvillian clearly changes at the $\mathcal{PT}$ symmetry breaking
point for an open 2-spin model with exactly balanced gain and loss if the total
spin is large. In particular, in a $\mathcal{PT}$ unbroken phase, some
eigenvalues are pure imaginary numbers while in a $\mathcal{PT}$ broken phase,
all the eigenvalues are real. From this result, it is analytically shown for an
open quantum system including quantum jumps that the dynamics in the long time
limit changes from an oscillatory to an overdamped behavior at the proposed
$\mathcal{PT}$ symmetry breaking point. Furthermore, we show a direct relation
between the criterion of Huber \textit{et al}. of Liouvillian $\mathcal{PT}$
symmetry and the dynamics of the physical quantities for quadratic bosonic
systems. Our results support the validity of the proposed criterion of
Liouvillian $\mathcal{PT}$ symmetry.
