'Pastie has it correct above, and it cannot be overstated that the solid rivets are better at dealing with imperfect holes. It is very very rare in aviation where the cheapest solution is also the lightest and strongest and longest lasting. Old fashioned solid rivets are one of those examples.
If you want to use pull rivets (Avex, Cherry, Huck, etc. etc.) then you can do it, but you do need to put extra time and care into making sure you have perfectly drilled, reamed, and deburred holes. That extra time could be more trouble than learning how to use a rivet squeezer or a bucking bar. Really.
One of the Parasol guys in Canada did a website article about rivet diameter and how many rivets to use for each gusset, etc. Well worth reading.
Chuck Beeson designed the Parasol the way he did because the angle and gussets allowed you to use a hand squeezer, and not have to use a pneumatic gun. Once you get a hand squeezer set up correctly and with only a little practice, it really is simple. Same with the rivet gun and bucking bar too. The fear of learninghow to rivet is far far worse than actually learning to rivet. Trust me, I know this from experience.