The wizard's parade tunic is a tunic wore by the wizards of Twinsun. They are given to everyone who graduates from the School of Magic.
When Twinsen finishes the three tests of the school of magic, the rector tells him that he doesn't have any tunics left, but that he can go see Chedil Amiradoo to get one. Twinsen then, after finding Chedil Amiradoo, shows him his wizard's diploma to prove he is a wizard and buys the tunic for 50 kashes. Chedil also gives him a false beard for free that makes him look like an old wizard.
The talking Sups throughout Twinsun, seeing Twinsen in a wizard's robe, offer him to go to the Hacienda del Monte-Pelado. The Sup there offers him to go to Zeelich through the secret base in the Temple of Bù.
A few NPCs have special lines for Twinsen in the wizard's outfit. E.g. Miss Bloop compliments him for looking "cute" in it.
Sup Agents that are normally aggressive towards Twinsen do not recognize him in the wizard's disguise and do not attack him.