Increased exposure to STDs,increased risk of emotional abuse (most grown adults fail with poly relations),increased risk of physical abuse, especially with young men being hot headed and often hierarchy mindedincreased risk of sexual abuse, "you are already dating two guys, why dont you want a threesome/gangbang, dont act like this" risk of pregnancy with denial by both potential fathers, or dispute about who takes the father roleincreased risk to develop commitment issues (those are already prevalent enough in our times)
None of these have to manifest, but it is perfectly reasonable for a parent to have an instant gut feeling against this.
Being worried about such a set up, especially with teenagers, who are emotionally and physically underdeveloped, where things going wrong can have life long consequences (STDs, unwanted pregnancy, trauma from emotional/physical/sexual abuse), it is perfectly reasonable to be worried, without it having anything to do with puritanism or christian fascism. These consequences are already possible with a monogamous relationship and get more likely with poly relationships.
It should be the parental instinct to be worried about their teenagers getting romantically active and to watch out for their children, without directly preventing them from doing thing or forcing them into hiding. It is only logical that this instinct gets more worried if not one but two, or even more people that could be a threat to their child are involved.