I'm going to make small quilts with my parents' clothing. Of course, there are several different types of fabric involved. What would be the best way to mix these types of fabric and have the quilt sew together well and retain its shape?

I think the biggest problem will be the different weights or the instability of some types of fabrics. Perhaps using a light fabric (muslin) for a foundation and either use a crazy or string design. It will be easier to stitch and will stabilise the fabric.

Hmmmm.......... first thing that comes to mind is are you mixing cottons, wools, flannels, etc?? Could be very challenging.

It sounds like a neat idea- Good Luck! Post pics!

since using old clothes, the only real issue is the stability.
It would be best to foundation piece them for sure.

After that, quilting will be an issue if you machine quilt. the different textures will not quilt the same, and that can cause issues.