Don't drag out the story of Meng's mother moving three times to justify relocating for your kid's private-school exams.

Literal

Meng-mother-three-moves [possessive-の] teaching [etc.-など] bring-out, child [possessive-の] [honorific-お]exam [at-での] moving [object-を] justify-[negative-do-ないでよ].

A sharp sentence. 孟母三遷の教え is a classical four-character reference to Meng Zi's mother moving three times to find a better environment for her son's education — a Confucian paragon of parental devotion. The speaker's complaint is that the listener is self-servingly invoking this hallowed tradition to justify modern お受験 ('private school entrance exams') social climbing. お受験 itself is an ironic honorific — attaching お politely to 受験 to mock the upper-middle-class obsession with prestige schools for young children. ~ないでよ is the negative request form with よ for emphasis.