I don't think driving for two hours is stress free. I'd take a shift stacking shelves over driving a 4-hour round trip any day.
Have you been a shelf stacker? Because it's hard work and it leaves you tired, grubby/dirty and feeling like you need a good scrub. Once, I was working in a far part of our warehouse and when I showered that night, when I washed my hair the water ran grey from all the dust and dirt. Also I would assume the DDs shift was at least as long as, if not longer than the drive time (without the sitting and socialising in between) .... in my supermarket the shortest shop floor shifts are 4 hours but most are between 6-8 hours.
Driving is tiring but in no way does it make you dirty ... and depending on the area you live 2 hrs could be stressful or it could be a pretty straightforward trip - I live in the Lakes and a 2hr trip could easily be a leisure drive through the country, we are a 45 minute drive from even the closest motorway, and it was his choice to make that trip.
When it comes to showering, I would give priority to the person who needed the shower - ie the one who needed to shower to wash off the dirt/grime from their day ... in this case the DDs need was greater than her dad's who just wanted a shower to wind down and refresh after his trip.
I somehow suspect if it was a teenage boy talking to his mum like this the responses would be very different.
Not if the situation was the same - I would still say the person who gets priority is the one who needs it more, and in this case it's the child.
Then again anyone who resorts to petty arguments such as "My house, my rules" and "I pay the bills so I get first dibs" does sound more like the child here so ...