The Parity Principle states that

if $X\neq \emptyset$ is a set, then there is $\mathcal B\subseteq \mathcal P(X)$ such that whenever $a,b\in \mathcal P(X)$ with $a\mathbin\Delta b = \{x\}$ for some $x\in X$, then $\mathcal B$ contains exactly one of $a$, $b$.

The Axiom of Choice for $2$-element sets $\mathbf{C}_2$ is strictly weaker than the original Axiom of Choice, and $\mathbf{C}_2$ implies the Parity Principle.

Does $\mathbf{C}_2$ imply the Parity Principle?

The Parity Principle states that

if $X\neq \emptyset$ is a set, then there is $\mathcal B\subseteq \mathcal P(X)$ such that whenever $a,b\in \mathcal P(X)$ with $a\mathbin\Delta b = \{x\}$ for some $x\in X$, then $\mathcal B$ contains exactly one of $a$, $b$.

The Axiom of Choice for $2$-element sets $\mathbf{C}_2$ is strictly weaker than the original Axiom of Choice, and $\mathbf{C}_2$ implies the Parity Principle.

Does the Parity Principle imply $\mathbf{C}_2$?