No. That ends poison. It doesn't give you a new save.
Lessor Restoration ends one condition, that condition can be poisoned, paralyzed, blinded or deafened.
If a creature is poisoned by a Carrion Crawler and you use it to end the Paralyzed condition the creature can succeed at the save. It is not a "new" save it is the same save he gets at the end of his turn anyway, the only difference is it is possible to pass.
If you use Lessor Restoration to end poisoned then you the poison condition would end and you would not need to save.
If you use Lessor Restoration to end Blinded or Deafened on a creature that is poisoned by a Carrion Crawler then that creature would continue to get a save at the end of their turn and they would automatically fail that save if they were Paralyzed, Stunned or Unconscious.
In no case is there a "new save".
All of which do not grant more saves.
No RAW you get a DEX save at the end of every turn whether you are Paralyzed or not. Nothing gives you more saves, they just enable you to succeed at the save.
I hope your many other examples are actual examples and not failed examples like those.
There is nothing failed about my examples. All of them can end the Paralyzed condition and they can end it on a creature that is Poisoned by a Carrion Crawlers.
I can give you other examples too - Any of the many creatures with Legendary Resistance can choose to succeed after automatically failing the save due to being Paralyzed.
Polymorphing the victim into a Giant Crab would make it immune to Paralyzation, enabling a possible sucsessful save.
Also not relevant. It would take an extreme corner case to allow a save while paralyzed, so the language they use means in English, the language they use, that you get specific save that overcomes the general paralysis rules.
What do you mean allow a save while Paralyzed? Where do you get this? Being Paralyzed does not take away the save you make at the end of your turn.
They always get a save whether Paralyzed or not, they just automatically fail it if they are Paralyzed.
I think this is the main problem here - you seem to think that automatically failing a save is the same as not making a save.
You have yet to show how that specific RAW is overcome by the general paralysis RAW in defiance of specific beat general RAW. Why do you think that general beats specific in this situation?
What specific RAW? There is no written rule stating that you can pass the Dex save you make when poisoned by a Carrion Crawler prior to a minute. You are making this up when saying this specific RAW exists.
The specific rule stating the save suceeds after a minute does overide the general rule for being Paralyzed and this rule would aply for other conditions too, includin Unconscious or Stunned. There is no such specific rule in the wording before 1 minute though.