I tried several games with the Parliamentary republics form and I encounter few issues that I'd like to share here.
But before, thank you for the mod, well done!

- [Issue] When a Theocracy is part of the republic, it is often part of the leading families (because it is often held by old dudes => high respect), which unfortunately lead to them being voted as the next Prime minister. Thus, destroying the republic (change gov of the realm into a theocracy)
I didn't try, but it is possible that a similar issue happen if a different form of republic (senate, concil, elected ruler) is voted to lead the Parliamentary republics. It may mess up the republic system.

- [Bug] Previous high respected Families tend to disappear in the list/senate for no reason, even if they are still part of the realm (often replaced by unlanded npc, the ones generated when there is not enough family).
This always happening in my game; I create a republic, I lose a vote as a prime minister, then after 1-2 years my families vanish from the senate/families, even though I am part on the strongest with some time twice respect of all the other families.

- [Bug] When the law "Citizenship Rights" is set to "Leader's Discretion", the next Citizenship law voted never happen and cancel/stuck all the following votes (therefore breaking all laws voting system of the republic).
Cannot be fixed in game other then changing the government with command (eg effect = { change_government = feudal_government }) and remake the republic from scratch. Or Attempt a coup also work, but it is a pain in the ass. :D

- [Balance] NPC Elected prime minister are often hated by their vassals, which often lead to the fall of the republic in the next few years following the vote. Only respect based vote seems note working well in term of balance of the game.

- [Exploit] With the decision "Attempt a coup". There is a way to cheese the game to spam an infinity quantity of army from this decision.
If you manage to hold a realm without landed vassal, (take Venezia for example, just remove any landed lords from the city/castle) then farm enough prestige to be Illustrious, then use Attempt a coup, then Convert to a Republic, etc...
A simple fix would be to limit only 1 Attempt a coup per life.
I don't think it is necessary to limit the decision Convert to a Republic (and I wish it won't be limited, it wouldn't make sense)

- [Minor issue] House of Commons counties shown control higher than 100 when hold with Absolute Control perk. No impact.

- [Minor issue] Missing text/info for citizenship laws vote (what is voted and what result is sometime confusing when there are a lot of votes happening)

- [Balance] It is more a suggestion than an issue and may be painful to implement, but I wish the elected prime minister could keep the crown authority of the previous leader.
It may help the realm to remain intact when held by a NPC.
Also, when the player/npc use the decision to Convert to a Republic, the prerequisite is to have at least a High Crown Authority, so it would make sense to keep it.
I understand it is a lot of work to be done when the leader change, but still wanted to ask this.

That's it for today, Thanks!