Sorry, but your Dad is probably being a bit too careful.
You're never going to kill a computer with static by touching its keyboard, mouse, or case.
Static damage does not affect well designed, modern electronics devices (including computers). Perhaps in the 80's - but not now.
Since my Degree in Electronics Engineering, I've killed probably 30 Integrated Circuit (IC) chips by unintentionally zapping them with rogue static charge on my fingers. However, once chips are mounted on a well designed board, the ground plane helps to make the entire circuit more robust, and the case helps to protect the entire product. In addition, IC manufacturers have been getting steadily better since the 70s at adding internal static protection circuits to avoid this very problem.
- You can ignore static damage if you're a home computer user. Modern electronic devices are so well designed that its not a problem any more (although it was up to the 80s).
- You must take care to avoid static damage if you're an electronics manufacturer, home electronics hobbyist (e.g. soldering ICs to a board), are installing computer parts (especially RAM or hard drives), or you are doing something super critical like installing a pace maker.