We present new radial velocity and trigonometric parallax measurements indicating that the unusually red and photometrically variable T2 dwarf 2MASS J13243553+6358281 is a member of the young (~150 Myr) AB Doradus moving group based on its space velocity. We estimate its model-dependent mass in the range 11-12 $M_{rm Jup}$ at the age of the AB Doradus moving group, and its trigonometric parallax distance of 12.7 $pm$ 1.5 pc makes it one of the nearest known isolated planetary-mass objects. The unusually red continuum of 2MASS J13243553+6358281 in the near-infrared was previously suspected to be caused by an unresolved L+T brown dwarf binary, although it was never observed with high-spatial resolution imaging. This new evidence of youth suggests that a low surface gravity may be sufficient to explain this peculiar feature. Using the new parallax we find that its absolute $J$-band magnitude is ~0.4 mag fainter than equivalent-type field brown dwarfs, suggesting that the binary hypothesis is unlikely. The fundamental properties of 2MASS J13243553+6358281 follow the spectral type sequence of other known high-likelihood members of the AB Doradus moving group. The effective temperature of 2MASS J13243553+6358281 provides the first precise constraint on the L/T transition at a known young age, and indicates that it happens at a temperature of ~1150 K at ~150 Myr, compared to ~1250 K for field brown dwarfs.
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