Chasing a modest 195, built around Srimantha Wijeratne's unbeaten 103, Nepal were staring down the barrel at one stage having lost nine wickets for 144.
However, the last-wicket pair of Karan KC (42) and Sandeep Lamichhane (5) added 51 off the final 47 balls to pull off a victory.
Parvez then bowled a wide, equalling the scores. With one needed from the final ball, Karan managed a single to seal a memorable win.
Earlier, Wijeratne's maiden List A hundred helped Canada post a modest 194 for eight as Nepal spinners Basant Regmi (3/34) and Lamichhane (2/40) kept the opposition batsmen under a tight leash.
The win was Nepal's second last-wicket heist of the tournament, and took them to four wins in five matches on the points table.
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