In an ordered sequence of $n$ random variables $X_1, \dots, X_n$, a smallest subset of variables $PA_j \subset \{X_1, \dots, X_j \}$ is called the Markovian parents of $X_j$ if $P(x_j | x_1, \dots, x_{j-1}) = P(x_j | pa_j)$, for all values $x_j$ and all values $x_1, \dots, x_{j-1}$ such that $P(x_1, \dots, x_{j-1}) > 0$.
In this definition $PA_j$ is not required to be unique. E.g. we could have both $P(x_3 | x_1, x_2) = P(x_3 | x_1)$ and $P(x_3 | x_1, x_2) = P(x_3 | x_2)$, in which case $PA_3 = X_1$ and $PA_3 = X_2$.
Pearl (Causality, 2009) says that the set $PA_j$ is unique if $P$ is a strictly positive distribution function. I.e. $P(x_1, \dots, x_n) > 0$ for all values $x_1, \dots, x_n$.
Can you give a sketch of the proof for why $PA_j$ is unique?