Green’s Dictionary of Slang

(US) the penis.

c.1885		in Randolph & Legman Ozark Folksongs and Folklore (1992) II 738: This was common in rural Missouri about 1885: You lie, you link, you fart, you stink, / You suck your daddy’s poodle-dink.
1992		(con. 1920–57) Randolph & Legman Ozark Folksongs and Folklore II 787: Other common names for the male organ are [...] pood, poodle, poodle-dink [etc.].