So, in a few words, the Parrens are basicaly the biggest pieces of sh*t, the Westerlands have ever seen, its what you get if you combine, the stupidity, cowaedness and greed of the Freys, with the anger and vengefullness of the Lannisters. So they are basicaly arseholes. Thats for the character.
Now comes the backstory.
The Parrens were the result of the union, between a Andal noble and an firts man noblewoman, related to House Reyne, which can be seen from the red and white colours on the Banner. The Andal knights brother, might have been the founder of House Jast, which can be seen from the inverted colours of the lionhead part, in their banner, (the Parrens have two black lionsheads on a field of yellow, and the Jasts, have a three yellow lionsheads on a field of Black, which also might mean that the Jast brother, was the thirdborn and the Parren brother was the secondborn)
The area the Parrens rule has a very vivid firts man presence, with a lot of weirwood stumps, standing stones and mystical graves of giants and ancient heroes. So as it was expected, the new lord Parren, and his succesors cut done those weirwoods, brought down the standing stones, or carved seven pointed stars on them, and raided the many different tombs, and build a great many septs and a septry, and even a small sept inside some ancient tomb on the mountains. And ofcourse they occupied the old first man keep, standing on a central rock formation, which almost stands in the middle of the valley, giving a central view to the owner of his estates.
As it may be expected for a House in the Westerlands, it is only fields and woods that they posses, but also a great number of mines, as the mountains and caves in that area , are full of precious metals and gemstones making it one of the richest areas in Westeros, and the lands of House Parren, are no different, filled with mines, the Parrens became rich easily, from all that silver that came out of those tunnels, and with their wealtg expanded their castle, to that monstrocity, that now stands, Seventowers, named from its seven grand towers that make it, (if you dont count the gatehouse) with tall and thick walls that fortress has been deemed impregnable by many, but in truth has fallen to a lot of hands in the years, as the infighting between the members of the family is constant and each one of them, wants to be lord. And ofcourse all that infightings takes a toll on the peasantry, as they regurarly see their homes burned, their crops stolen and sons killed to serve each lords will. That attitude has ofcourse, as a result, a great number of revolts, which always almost end in bloodshed and heads on spikes on the castle walls.
Once that infighting went so far, that one of the two twin brothers who fought for the lordship, went to ask the help of the King of the Rock with his army to bring down his brother, and make him lord. He ofcourse suceeded, but at what cost, as the grand tower of the guarrison in the castle crumbled on the cliffs below, killing many. His reign btw, was a peacefull and prosperous one, as he actually somehow cared for his serfs, but still he wouldnt wait to hang them for any petty crime or reason.
In the years after the Conquest, the Parrens continued their bullshitery and took parts in some major events ( see the Canon stuff) and then at some point, after the tourney at Ashford meadow, maybe during the Great Spring Sickness, the main branch of the family went extinct, they were then succeded by their Cadet Branch, which was then living in Lannisport, continuing the Parren name and arseholeness that comes with it. ( it is only semi canon that they are extinct, so I decided to change that fact, a bit, and made it half true, as the main branch is dead, and they are ruled by a Cadet one)
Regarding their relations with other Houses, I cant say they are the best, as for their attitude most nobles hate them. Here follows a list of a few prominent Houses in the Westerlands and the reasons they hate the Parrens:
House Lannister: The Parrens most of the time, don't deliver all the tax they are supposed to. It is suspected they always hold a good sum of the silver. They blame the bandits that plague the northern pass to Lannisport, even though after their appearance, the carriages pass with an escort of Lannister riders through Crakehall, without at all passing close from the holdings of the bandits.
They also sometimes say that the bandits stole it from their castle, which is nigh impossible, as that monstrocity perched on a high cliff, with thick walls and high towers and only one way in an out, which is protected by a huge gatehouse, and a massive door made of Ironwood reinforced by massive bronze plates, and steel joints, making it almost impossible to break into.
House Crakehall: Parren men are occasionaly found poaching in the woods surrounding Crakehall, hunting the lords boars and stags and cutting their trees for firewood.
House Swyft: The Swyfts were for a small period vassals of House Parren, back in the day. The Parrens, with their supposed higher status in addition to their neighbors, ussualy bullied them, and asked for way more tax, in silver, than they could afford. But after fighting against king Maegor, the Parrens lost all of their vassals, including the Hetherspoons which they treated in a similar way.
House Clegane: The Cleganes gained their land, from lord Tytos Lannister, who hearing his councelors advice, took lands from the deceatfull and disloyal Parrens, which then were a close ally to the Reynes and Tarbecks(and if not for the lady Parren, the infamous rebellion would have been called, the Reyne, Tarbeck and Parren rebellion.) Since then, the Parrens scold and bully the Claganes, but have stopped since ser Gregor became the Knight of Clegane's Keep, which was a former Parren Holdfast.
House Banefort: A lord Parren, once crippled the heir of lord Banefort in a melee, by hitting him hard with his morning star on the neck, crippling the poor youngman, who then died a day after, as he was not able to breath. Its said that the victor spat and laughed on the poor child while he fell to the ground.
House Marbrand: A lord Parren once attended a banquet held by lord Marbrand, in his castle of Ashemark. The lord Parren got so drunk and heavy from all the food he ate, that he scolded on the host inside his own home, with the guests, vomited a few times in the great hall almost ended up getting in bed with lord Marbrands daughter. He also was said to act as if Ashemark was his own home and yelled at the poor servants. The lord Marbrand, after the banquet went to Casterly rock to see the Lord Lannister, who was unable to attend due to some health problems, regarding lord Parrens unfitting behaviour. The Marbrands still argue with the Parrens about that night, even though most arguments ended up in fighting between them. As most of the time the Parrens feel no shame, and will easily lie and defend their views even though evidence say otherwise.
House Sarsfield: A lord Sarsfield once attended a celebration at Seventowers. Due to his history with the lord Parren, as he was said to owe him 30 golden dragons from a gamble they once had on an outcome of a tourney, was said to have been gotten drunk by the lord Parren and then thrown off the battlement on the cliffs below, and was impaled on a big pointy rock. The Parrens deny the alligations that he was murdered, and say that he was just too drunk to stand and fell of the walls. The families still argue that matter to this day.
House Vikary:Once a lord Parren attended a banquet at Castemere hosted by lord Reyne, a guest there was the knight of Vikary. After lord Parren got horribly drunk, he insulted the knight of Vikary countless times, for his lack of lands and riches and for his bastard backround, calling him unworthy to be noble, ofcourse, the Vikarys got quite a bit pissed off by that actions and they still are quite pissed off with that night, ofcourse the Parrens deny everything, and there he have one more family angry with them.