You take the poison damage regardless
The Yuan-ti Pureblood's shortbow attack states:
Hit: 4 (1d6 + 1) piercing damage plus 7 (2d6) poison damage.
Which means that you take the poison damage on a hit, regardless of whether you take piercing damage.
Crawford answered a similar question on Twitter:
Q: If the Battlemaster maneuver parry, reduce the weapon Drow damage to zero, the poison damage still work?
A: Drow poison in the DMG is delivered by piercing/slashing damage (0 dmg = 0 poison). Poison in the MM's drow is delivered by hitting.
Many drow have an attack with the exact same wording as the Yuan-ti Pureblood's shortbow attack, for example the Drow Elite Warrior's shortsword attack:
Hit: 7 (1d6 + 4) piercing damage plus 10 (3d6) poison damage.
Injury poison does nothing if the piercing damage is 0
There are poisons that behave the way you expected. These are injury poisons and are explained on page 257 of the Dungeon Master's Guide:
Injury poison can be applied to weapons, ammunition, trap components, and other objects that deal piercing or slashing damage and remains potent until delivered through a wound or washed off. A creature that takes piercing or slashing damage from an object coated with the poison is exposed to its effects.
However, these rules are meant for players and don't seem to apply to monsters. For example, the "drow poison in the DMG" which Crawford mentioned is an injury poison, which means piercing or slashing damage must occur for the poison to take effect, yet many drow have a hand crossbow attack that delivers an effect identical to the drow poison on a hit, regardless of the piercing damage.