I would fly into LA and rent a car, do LA for a day or 2 (I'm not a fan of cities) then drive to Vegas, maybe with a stop in Death Valley and stay a night or two in Vegas. Next drive to Grand Canyon for a night or two, head for Phoenix with perhaps a night in Sedona. Turn in the car after you've seen Phoenix (there will be drop off fees).

Fly to San Francisco and rent a car to visit Yosemite then return to SF and turn in the car and use public transportation in the city.

The trip could be done in the opposite direction also.

For Grand Canyon, get lodging in the park. The nicest and most expensive is elTovar, but Kachina, Thunderbird and Bright Angel are right on the rim too and Maswick and Yavapai are just a short walk back. They are basic motel type places, except elTovar, clean but no frills.

Yosemite will only have the valley open at that time of year and they should check the road conditions before they venture out as there might be snow or even road closures at that time of year so making this near the end of the trip lessens the snow possibility.

From SF - they could drive the 4 hours to Yosemite and then if they have the time and the inclination - drive from Yosemite over to the beautiful Coast hiway/1 - and stop at Carmel/Monterrey for a night, and then enjoy the super scenic drive through Big Sur, etc.

Due to winter snow - you can't get out the east end of Yosemite - through Tuolumne Meadows/Tioga Pass to go down 395 to Vegas - through Death Valley. BTW - not a big fan of Death Valley - especially not to make a special trip there.

In LA - love the Getty Museum and most like to take a quick tour of Hollywood, etc.

From LA - it's about 4 hours driving time to Las Vegas - and they can then drive 5 hours to the Grand Canyon, and/or 3 hours to Zion, etc.

And for my additional two cents - forget Phoenix and kick back down here in San Diego - which has less congestion than LA.

If they are OK with twisting roads, one interesting route would take them on I-80 to Placerville and the State Route 49 shouth through the Gold Rush Country to catch SR 120 into Yosemite at the Big Oak Flat entrance. Since the road over Tioga pass will be closed, go into Yosemite Valley then when exiting take SR 41 into Fresno, then SR 99 to Bakersfield, SR 58 to I-15 and LAs Vegas. If they want a Death Valley side trip, leave I-15 at Bakerfor SR 127 to SR 190 in Death Valley. You can avoid backtracking by going to Las Vegas via Parhump. Then from LV it's US 93 over the new bridge at Hoover Dam into Kingman thence I-40 to Williams for Death Valley. You could visiti Sedona and Jerome before heading to Phoenix or skip Phoenix and go to San Diego and then LA.
Except for Disneyland area, I think the Universal City area is a good base from which to see LA. Just remember traffic can be a real time consumer.
From LA back to SF along US 101 as far as San Lusi Obispo and then Highway 1 to Monterey.
Long trips, lots of driving, but doable. Possible other stops, Sonora, Death Valley, LV, San Diego, Santa Barbara, Cambria. Hearst Castle, Big Sur, Half Moon Bay.

Do they have reservations for Yosemite and Grand Canyon to stay in the park? If they want to stay inside the park you need reservations, keep calling every day as folks do cancel.
Another thing to keep in mind, there may be snow in or on the road to Yosemite. If the Highway Patrol imposes requirements for chains on the cars, most rental car agencies do not allow the use of snow chains.
I agree with dropping Phoenix, boring city. Go to San Diego instead, really nice town with lots to see and do there.

We did part of this trip in early January. Flew into Phoenix, stayed a night in Sedona, two at the Kachina Lodge GC, then drove to Las Vegas via bits of Route 66 and then on to LA. Great scenery, loved Phoenix and Sedona & the Canyon. Vegas was too 'fake' for me but I'm glad I saw it. LA is just a big, smoggy city but I'm glad I went there too.
You get read our impressions on my trip report on here (it's long and takes ages to load) or on my travel blog.
http://demansersintheusa.blogspot.com.au/2012/12/discovering-desert.html

Your parents will love this part of the country