When casting Silvery Barbs, could one apply the Order Domain class feature Voice of Authority to the second half of the spell?
Yes, this works.
Voice of Authority states:
If you cast a spell with a spell slot of 1st level or higher and target an ally with the spell, that ally can use their reaction immediately after the spell to make one weapon attack against a creature of your choice that you can see.
Silvery barbs is a 1st level spell, and says:
You can then choose a different creature you can see within range (you can choose yourself). The chosen creature has advantage on the next attack roll, ability check, or saving throw it makes within 1 minute.
If you select an ally for this part of the spell, you have met all the requirements for using Voice of Authority (cast a 1st level spell, target an ally), and your ally may immediately use their reaction to make one weapon attack with advantage. The attack is with advantage because Voice of Authority states that the reaction happens after the spell.