Lair Actions are used in combat, and are not available while the creature is incapacitated or otherwise unable to take actions.
From the introduction to the Monster Manual:
If a legendary creature has lair actions, it can use them to harness the ambient magic in its lair. On initiative count 20 (losing all initiative ties), it can use one of its lair action options. It can’t do so while incapacitated or otherwise unable to take actions. If surprised, it can’t use one until after its first turn in the combat.
So lair actions are used on initiative count 20 - this means we are still in combat when lair actions are being used. How are you going to keep track of lair actions and initiative for the duration of a short rest? That's 600 rounds of combat. The rules permit the use of lair actions only on initiative count 20, so they cannot be used unless we are tracking initiative.
It doesn't work anyway: let's check this rule against the rules for short rests and long rests. The rules for short rests state:
A short rest is a period of downtime, at least 1 hour long, during which a character does nothing more strenuous than eating, drinking, reading, and tending to wounds.
If the aboleth is using its lair actions, it is harnessing the magic of its lair to engage in combat with the player characters. This is not consistent with the description of a short rest.
The rules for long rests state:
A long rest is a period of extended downtime, at least 8 hours long, during which a character sleeps for at least 6 hours and performs no more than 2 hours of light activity, such as reading, talking, eating, or standing watch. If the rest is interrupted by a period of strenuous activity — at least 1 hour of walking, fighting, casting spells, or similar adventuring activity — the characters must begin the rest again to gain any benefit from it.
If taking a long rest, the aboleth needs to sleep for six hours, during which it is incapacitated1, and unable to take actions. It can awake for two hours of the long rest but if at least one hour of those two is spent using lair actions on the players, then the long rest is lost.
But it doesn't matter in this particular case, as the aboleth's Lair Actions require line of sight on the targets or close proximity to the aboleth.
In this case, if the aboleth is able to use its lair actions to engage the party, the party is close enough to engage the aboleth. The aboleth's lair actions require either direct line of sight within 60 or 90 feet, or the aboleth to be at most 110 feet away from the target. If the aboleth is that close, no one is resting, not the party, and not the aboleth.
1 This is technically an optional rule found in Xanathar's Guide, but generally speaking, being asleep should deprive you of your ability to take actions, even if it is not explicitly stated to be so in the Player's Handbook.