When approached by a friend-become-suitor dad should hear him out graciously, take a few days, pray and talk with his wife and daughter. If the answer is yes, he spends time with the family, together. If that goes well, they have time more alone, still accountable, maybe completely by themselves, depending. If things start deteriorating, the father should intervene pronto, and show him the door. If that goes well, the young man should be asking for her hand shortly. If he doesn't, the father should call the question, discerning whether it is a cold feet "yes," or an indecisive "no."
Don't spend so much time and energy on the wedding - the doorway into the new household - that you neglect or trash the new household itself. At a wedding the family reproduces itself, while the church and state witness it, binding the new household to covenant faithfulness.
Marriage brings out the selfishness or servanthood in a person. Honor and serve each other out of honor for God.
Objection to courtship #1: parents can make mistakes
Answer: so do the young people; the Bible puts dad in charge.
Objection #2: how can we require what we didn't do ourselves?
The Bible sets the pattern, not our experience. Do you permit sexual impurity in your children if you weren't pure yourselves?
Objection #3: how can we get to know the young people that will be interested in our children?
Have them in your home in a group setting, that doesn't pair people up. Character is revealed far more clearly in a group setting than in one-on-one dating. Watch how he talks to his mother, more than how he smooth-talks you on a date. The couple should not be growing attached while putting up appearances with each other on dates, before discerning each other's character more objectively in group settings. Relational hyper-sensitivity and angst in young people ("she likes me! oooohhhh, she doesn't like me...) needs to be matured by parents who encourage and admonish - it won't be cured by courtship.
Pairing off informally is bad news. It leaves both parties, especially the girl, unprotected. You need to commit to courting a girl with the intent of marrying her, after discerning her character is good and you are attracted to her, but before you have any intimate knowledge of her outside a group.
Jealousy is bad news... jealousy among the parents!
It is disguised, because it is concern for another: their child - but it is still sin to be jealous for your child against another, where there is no marriage covenant to be rightly jealous over.