It is a little more complex than the arguments above.
When you crank the engine to start, the battery provides a lot of wattage for a few seconds.
Up to 100 Amps of cranking current @ 12V (12V*100A = 1200VA or ~1.2 kW). (disclaimer numbers are WAG - wild ass guesses)
If it takes 3 seconds to crank, then that is 3.6kWs of energy or 3600 Joules.
Of course the alternator carries the entire car's electrical load once the car is started AND recharges the battery.
But the battery will not accept a charge in 3 seconds like it took to discharge it, it has a charge current that depends on a lot of things like input voltage, temperature, and age of battery.
What is the charge rate of the battery right after starting?

, but if it is 10 amps, then it should only take 36 seconds to recharge.
1 Amp of charge would need 360 seconds or 6 minutes.
Turning on all of the car's systems lowers the system voltage which will reduce the battery charge current, which would take longer than leaving all systems off.
This only has a small effect IF the alt. is running under capacity as the output will remain in regulation.
So - it depends.
My BRZ definitely did not like taking lots of short trips and it WOULD eventually drain the battery and not start. It was worse during the winter (cold battery = slower charge?).
I had to replace the POS OEM battery at 35k miles.
So your dad is right - Lots of very short trips means you are slowly draining the battery by not letting it fully recharge each time.
What is the length of time it takes to get the battery back up?

But turning on all of the car'e electrical systems right after starting COULD mean it will take longer to fully recharge.