Political Parties and Their Use

Depends on what you are trying to do in game and what your current situation is.  Every political party offers certain useful bonuses simply for being the party in power, plus additional optional laws available when their power reaches certain thresholds.  Their base perks being felt even from the beginning of the game.

Here's an example.  Let's say you are in a war.  You have the reseach to extract adamantian and you need adamantian to build certain powerful ships for your war effort.  But you don't have any adamantian on any of your colonized planets, but one of your systems does have a deposit on a planet not currently colonizable by you, because you haven't researched the required tech yet.  But right now your research is tied up with other tasks you need more immediately for various things and the research you need to colonize said planet is 2 or 3 techs at least from being achievable, not including the turns required on top of that.  But what you do have, is an election.  

Well you have a couple options available.  You can either elect the Sciencists party, which would potentially unlock the next tier of research up (the Scientists base perk) allowing you to research the required tech earlier, allowing you to colonize the planet you need with the required adamantian.  Plus the Scientists have an excellent low level law you can enact called the "Dirty Hands Act", which reduces the production cost requirements for all system improvements by 10%.  Which could further help you by reducing the number of turns required to build any structure.  So Construction, Science, Dust, Influence, Morale, Wonders, all buildings would be considerably quicker to build on all planets, allowing you to significantly improve your economy more quickly.  So this law could prove very beneficial to factions that struggle with production, such as the Sophons.  Especially early game.

Or you could elect the Ecologists party, whose base perk makes all non-gas giant planets colonizable regardless of whether you have the tech or not.  There is a bit of a FIDSI penalty for colonizing planets you don't have the research for yet, but it opens up 80%-90% of your uncolonized planets to colonization and even small bonuses to your FIDSI by colonizing these planets is better than no bonus at all.  Especially when it opens up important resources to you that you otherwise would be locked out of.  Plus those penalties will disappear once you complete the required research for those planets.  So if you were to use this to your advantage, it could prove to be an even bigger improvement to your situation than if you had chosen the Scientists party.

I couldn't possibly list every perk or law for every political party.  But every political party has a unique and useful base perk that you should pay attention to and use to your advantage to suit your needs at any given point in a game.  And every political party has unique laws that will benefit you throughout a game.  Some political parties even sometimes bring much needed benefits to specific factions, such as the Religious party to the Vodyani where they benefit greatly from the Religious party's low level law which gives you a base level of essence per pop, per turn (especially early game).  Which is required by the Vodyani in order to build Arks and help populate their Arks more quickly.  And even mid-late game with the "Sinners and Saints" law which locks your morale at "content".  Which really helps the Vodyani because they tend to have significant issues with low morale mid to late game, and also allows them to expand unhindered by the overcolonization penalty.  So you will see a lot these kinds of benefits throughout the political parties.  But it's up to you to use them to your advantage.

Hope this helps.

Access to laws you want/need on demand + completing quests which demand specific political party/amount of laws running.

Is what it comes down to.

Bear in mind that reaching higher tier laws may require multiple election cycles and that even party with lowest preference can randomly win the elections regardless of your attemps (unless autocracy/dictatorship).

As mentioned the example "saints and sinners" is very a influential law and acts as a baseline for craver players not only on larger maps effectively ignoring dissaproval mechanic and stacking slaves without penalties. With that in mind, as a craver, you want to continuously select religious party so that you may reach it ASAP.