That is the Dragonback series by Timothy Zahn. The first book is called Dragon and Thief and here is an excerpt as per Goodreads.
Jack Morgan 13 has a dragon on his back - literally. The warrior K'da species are symbiotes, can only live six hours without their humanoid host. Draycos 30s is the sole survivor of the three ships destroyed scouting for their refugees, and Jack is the only possible choice. They may be an odd couple, but together they're more than the sum of their parts. They are destined to travel far and wide, facing many perils - human, alien, and other, as they seek justice and safety - Jack for himself, and Draycos for his people
First published in February 2003.
The excerpt for the next book Dragon and Soldier mentions the tattoo bit.
Jack Morgan 14 aka Montana enlists with youngest trainees for Whinyard's Edge mercenaries, hired by Parprin daublite mining colony humans from "Agrist claim-jumpers", who in turn employ Shamshir mercs. Honorable warrior-poet Draycos 30ish hides as 2D tattoo on Jack's back, and encourages noble values while their space-ship Uncle Virge continues the tradition of his late selfish Uncle Virgil, con artist. Recruit Alison is expert, already has militia experience - and secrets. Somehow, the duo must track down who attacked Draycos' scout ships before the refugee ships arrive to be ambushed.