Depends on how long you want to take on the trip.
^^--- 100% this. That's a long trip, already most of a day's drive. Any time you spend running 15 mph on dirt roads vs 75 mph on the Freeway will take 5x as long, plus the fact that the dirt roads require more driving distance because of curves and hills.
I have a similar length trip to see my folks (~5.5 hours straight driving, plus potty and food stops) and I've investigated multiple diversion trips on Google Maps. Some day I'd love to take the Trans-America trail, which goes from close to here to close to there. But ultimately the goal is to get there to see Memaw and Papa, and my wife, kids, and dog would flat-out mutiny if the normal 7-hour trip took two days. So what I do is get there as quickly as possible, then at some point during the trip slip off and do a little driving down a few private hunting club roads I have access to.
Back to your situation...
The PCH is beautiful, but it'll stretch your trip out to about 9 hours.
One option would be to blast down 5 to the mountains just north of LA then divert along forest service roads. I found a route where you get off 5 and head a little east at Lancaster Rd (Hwy 138), then cut south over the mountains to rejoin 5 at Castaic. It adds about 1.5 hours to the trip. All maintained roads so no serious wheeling, but it won't be pavement.
Another option would be to go south to San Jose then follow Airline Hwy (St. Rte. 25) then Coalinga Rd. Rejoin 5 at Colinga. All paved, so more fun in a zippy sports car than an SUV, but different from the freeway. This adds about 2 hours.
Most of the trip is through private farm/grazing land. There are dirt roads parallel the freeway, but usually those will be private and gated. And even if the gate at one end is open, the one at the other may be closed.
If you were in SoCal (Riverside County) I could point you to countless utility service roads across BLM land (ungated), but I'm not really familiar with many areas north of LA. Maybe someone local can point you to an unmapped semi-parallel side route.
Good luck!