"How can I involve parents who are not Catholic in the preparation of their children for the celebrations of First Reconciliation and First Eucharist?"
Encourage parents to come to catechetical sessions before preparation of the children begins. If non-Catholic parents ask if they should come, assure them that they are most welcome. Do not try to separate out parents who are Catholic from others, but you might make a statement before beginning the session like, “I know some of you are not Catholic and might have some particular questions. I am happy to answer questions during the session or, if you prefer, I will be around after the session, or feel free to come and visit the office at another time so we can clarify anything you might be wondering about.”
Another way to involve everyone is to have parents come, a week or two before First Communion, for a retreat you might call Jesus Day. Have some stations for making bread, reading stories, and having families telling their own stories. At the end of the retreat, encourage everyone to eat the bread they have made and have a cup of grape juice. Also, be sure to tell parents that if they are celebrating as a family, the non-Catholic parent is welcome to accompany their child to the table and simply cross their arms so the priest will know to give them a blessing.