@Sallyg
If it were me. I'd definitely cut off that parsely stem and leave BST at home. Support stem on another plant or even crumpled paper or tissue or toilet paper roll so chrysallis hangs down in same position as shown in photo. I've taped and tied weak stems/brenches to another branch or a chop stick. You may need two sticks, one to reinforce parsely stem and another for botton of c to rest on. As best as I can recall, most of mine were perpendicular to the ground, not horizontal. My mental picture is a mountain climber in harness rappelling down a cliff face. I also think the chrysalis splits open along the back (away from support stick) for emergence, not like the Monarch from the bottom.
Also don't know whether yours will emerge this year or overwinter. I got some metalic thread from Joanns Fabric for my brood four years ago and use it with my Monarchs too.