It seems nobody has posted the original yet. How can you appreciate the parody if you don’t know the original? You can hear this at the Internet Archive:


THAT’S AMORE
Words by Jack Brooks, music by Harry Warren, ©1953
As recorded by Dean Martin, with Dick Stabile and His Orchestra, on Capitol 2589, 1953.
From the Paramount picture “The Caddy.”

In Napoli, where love is king,
When boy meets girl, here's what they sing:

When the moon hits your eye like a big pizza pie, that’s amore.
When the world seems to shine like you’ve had too much wine, that’s amore.
Bells will ring, ting-a-ling-a-ling, ting-a-ling-a-ling, and you’ll sing “Vita Bella!”
Hearts will play, tippy-tippy-tay, tippy-tippy-tay, like a gay tarantella.

When the stars make you drool just like pasta fazool, that’s amore.
When you dance down the street with a cloud at your feet, you’re in love.
When you walk in a dream but you know you’re not dreaming, signore,
Scusa me, but you see, back in old Napoli, that’s amore.

- - -
amore = love
vita bella = beautiful life. There is a song called "La vita è bella" (Life Is Beautiful) but I'm not sure that's what the songwriters had in mind.
tarantella = southern Italian folk dance in 6/8 time
pasta fazool = New York Italian dialect for pasta e fagioli, pasta and beans
signore = sir
scusa = excuse (but “excuse me” should be mi scusi)
Napoli = Naples