Yet there can be concerns. Children might regard the new sweetheart as a bad fit for their parent. Maybe the adult child can't get beyond their own sadness and accept their father being happy with someone other than their mother. And there are some adult children who see a parent's new relationship as a threat: That "new partner" is just after their money.
These are all legitimate concerns and it's prudent to be cautious, but at a certain point it is your partner's decision and should be respected. In terms of etiquette, we encourage you to focus on your parent's happiness and on keeping a positive relationship. "Mom, I like the man you introduced me to the other day, and I think he likes you. I want you to be happy. It would be wonderful if you started seeing each other... whenever you're ready of course!" Your parent will know when and if it feels like the right time to start dating.