Hi Joyce,

There is a solution for your problem that won't involve digging up your lawn, but will require careful application.  There are products on the market available at your local garden center and online, from at least two companies. 
Since you have disposed of the two parent trees, you won't need to worry about harming those which could have been a cause for concern.  There is some question about how effective they are as you can read in this fact sheet from Iowa State University:  https://yardandgarden.extension.iastate.edu/how-to/managing-suckers-fruit-ornamental-and-shade-trees

Utah State University has a comprehensive fact sheet that provides both the chemical and trade names for products that can help:  https://extension.usu.edu/productionhort/research/managing-suckers-around-fruit-trees

If you search online using the word suckerstopper, you can find commonly available products.

Best of luck.