Chasing an unlikely 364 for victory, WA looked gone for all money after slumping to 9-205 with 28.1 overs still remaining on day four and Paris (25 not out off 82 balls) coming in at No.11 with a runner, having injured his hamstring earlier in the match.
With wicketkeeper Joel Curtis (19no off 113 balls) at the other end, the pair blocked and blocked their way through every over - refusing to take even the easiest of runs.
It meant any scores they got came through boundaries.
As NSW's desperation grew, so did the number of fielders surrounding the batters.
When spinner Tanveer Sangha was bowling at the death, every single fielder was placed around the batter, making for a bizarre sight.
But it was all to no avail for NSW, who absolutely dominated the match but could not produce the goods when it came to dismissing the final batter.
The result continued an unlikely theme for WA, who beat Queensland by two wickets in their Shield opener when Sam Fanning, also nursing an injured hamstring, batted his team to a famous victory.
NSW have now posted two draws from their opening two games of the Shield season.
WA endured a disastrous end to day three, losing 3-1 as Cameron Bancroft (11), Sam Whiteman (17) and nightwatchman Corey Rocchiccioli (0) all departed in swift fashion.
That left the hosts at 3-30 by stumps and requiring a further 334 runs for victory - or surviving the entire fourth day to force a draw.
Nightwatchman Liam Haskett (17 off 101 balls) dug in brilliantly alongside Jayden Goodwin (67 off 165) to frustrate NSW's attack during the morning session.
Goodwin was left feeling crushed when he tickled a wayward Jack Edwards delivery down the leg side for an easy catch to wicketkeeper Ryan Hicks, leaving WA at 5-143 midway through the day.
WA's hopes of a draw were revived as debutant Corey Wasley (29), who was substituted into the game after skipper Aaron Hardie injured his left shoulder, and Hilton Cartwright (34) dug in for the next 12 overs.
But Ben Dwarshuis (4-50) and Sangha (2-43) snared two wickets apiece across the next eight overs to set up what seemed at that point to be an easy victory.
Paris, playing his 56th first-class match, and Curtis, making his 19th appearance, had other ideas, blocking WA to a draw which could prove crucial in the Warriors' hunt for a top-two berth.
NSW opener Sam Konstas was man of the match after producing knocks of 71 and 151no.
