The chances of getting hurt are slim as long as you are aware how to tackle properly and how to fall. Obviously there is still a risk, but wearing proper protection such as gum shield and body/headguard(optional). If you were really serious you could attempt to build stronger shoulders by doing exercises such as lateral pulldowns to increase shouldee strength and reduce injury risk. But overall, good technique will prevent the type of injury your dad is afraid of 9/10times.
I have played/watched highschool rugby for years, and I haven't seen a single person get injured legally. I have only seen a couple injuries, most of which are better in a couple days. Hell, the only injury on my team this year was a broken collar bone because somebody picked him up, spun him, slammed him into ground, and fall on top of him after (WHICH IS COMPLETELY ILLEGAL!!!!!).
If rugby is played properly no one should get hurt. But people do, do stupid stuff and can get other people first.