The story
From piss (“urine”) + proud. Appears in Francis Grose’s A Classical Dictionary of the Vulgar Tongue (1796): : “PISS-PROUD. Having a false erection. That old fellow thought he had an erection, but his — was only piss-proud; said of any old fellow who marries a young wife”.
Etymology from Wiktionary, CC BY-SA 4.0.