12. If sets $A_{1}$ and $A_{2}$ satisfy $A_{1} \cup A_{2}=A$, then we denote $\left[A_{1}, A_{2}\right]$ as a pair of subset partitions of $A$. It is stipulated that: $\left[A_{1}, A_{2}\right]$ and $\left[A_{2}, A_{1}\right]$ are considered the same pair of subset partitions of $A$. Given the set $A=\{1,2,3\}$. Then, the number of different pairs of subset partitions of $A$ is.
12. 14.

Divide into three categories.
First category: When at least one of $A_{1}$ and $A_{2}$ is $A$ (let's assume $A_{1}=A$), then $A_{2}$ can be any subset of $A$, for a total of 8 different subset decompositions.

Second category: $A_{1}$ and $A_{2}$ are one-element and two-element subsets, respectively, with the following three subset decompositions.
(1) $A_{1}=\{1\}, A_{2}=\{2,3\}$;
(2) $A_{1}=\{2\}, A_{2}=\{1,3\}$;
(3) $A_{1}=\{3\}, A_{2}=\{1,2\}$.
Third category: $A_{1}$ and $A_{2}$ are both two-element subsets, with the following three subset decompositions.
(1) $A_{1}=\{1,2\}, A_{2}=\{1,3\}$;
(2) $A_{1}=\{2,3\}, A_{2}=\{2,1\}$;
(3) $A_{1}=\{3,1\}, A_{2}=\{3,2\}$.
In summary, there are $8+3+3=14$ different subset decompositions.