In a nuclear reaction, both the total mass number ($A$) and the total atomic number ($Z$) are conserved. The given reaction is $X \rightarrow ^{14}_{7}\text{N} + 2\beta^-$. A $\beta^-$ particle is an electron, represented as $^0_{-1}e$.

1. Nuclear Equation: Let $X$ be represented as $^A_Z X$. The equation is: $^A_Z X \rightarrow ^{14}_{7}\text{N} + 2(^{0}_{-1}e)$

2. Conservation of Mass Number ($A$): $A = 14 + 2(0) = 14$

3. Conservation of Atomic Number ($Z$): $Z = 7 + 2(-1) = 7 - 2 = 5$

4. Number of Neutrons ($N$): The number of neutrons is calculated as $N = A - Z$. $N = 14 - 5 = 9$.

Therefore, the number of neutrons in the parent nucleus $X$ is 9.