He is Evelyn McCarthy-Parsons' arranged husband. This is a mutually beneficial relationship - they both want to consolidate power, each for their own reasons, and are willing to employ less than squeaky clean means to achieve it. They trust each other and are partners in crime, and one could say they're the best of friends. Their relationship turns sexual on occasion, but it's with no strings attached;
They primarily help each other if something happens rather than focus on their own power exclusively, being a very powerful team, and aren't inclined to break it apart anytime soon;
Publicly, his relationship with his life is very loving and they act the part. Everybody knows the Parsons are a very close and affectionate couple, and they're raising an adopted child together;
The Parsons are a pureblood family with a rich history, much like the McCarthy's. This makes it so the joining of both families is a particularly strong power play in the grand scheme of things;
He should be very caring with his daughter as well (Alyona Parsons), but in all likelihood he isn't nearly as close to her as Evelyn is;
He should hold some sort of important position within the Ministry of Magic. The department doesn't matter, but he should definitely be climbing the career ladder just like his wife is for the sake of power;
He most likely has affairs outside of his marriage, but he's careful with them. He's aware that Evelyn does the same, and it shouldn't bother him at all;
He is aware that his wife is barren;
From marriage to his wife, he's also related to Amber Cross, Gemini Demetra, Samantha & Keira Hammond (all are her cousins in various degrees), and Sibylla Townsend, her aunt (as well as the whole range of Hammond adoptables out there + Mairead Townsend). Have an over-complicated family tree. Currently, the connection to Isolda Richards is unknown, but she'd be off limits from their schemes regardless due to a genuine friendship between his wife and the woman;
Outside of that, personality/history/reasons behind his lust for power/extended family/play bys/etc are all up to you. He should be his own thing and not background for Evelyn's plot.
As someone with a large array of connections and plans to become the Head of Magical Commerce & Business (as well as aid his wife in becoming the Head of Magical Law Enforcement), he most likely has a storied history with various folks that he should expand on.
Him and his wife are quite close in a number of ways, even if their marriage distinctly lacks the 'romantic' component to it (although the physical component is sometimes present). They go to each other when they need to vent or let loose. It's definitely something that should continue.
What are Patrick and Evelyn planning next? What will get them to the next level? Who and what should they target, and how should they go about it?
He and his wife need to put up a good front and look like they're happily married. A few dates wouldn't hurt too much now, would they?
How does Patrick get along with the rest of the family? Those are a lot of Veela women. Can he truly fool them all?