Thing is she gets bullied on the play ground.
What behavior tells you this besides the fact that she lies on her back around new dogs.
What was she like during her heat. Was she in any way defensive then?
if there's not a way to boost her confidence so she's not so fearful or submissive.
Again what other evidence is there that she is 'fearful or submissive'.
I would think by actually lying down other dogs can't bully her. She is avoiding conflict by acting in a submissive way but maybe that is her technique until she is stronger.
You just let her off in a dog park to fend for herself and wonder how to build confidence and why she is acting submissive to a pack. She is defending herself and lowering herself to potential threats to avoid conflict. I'm not sure if that is a bad response or a fearful response based on the info you gave.
Does she show stress or tension or is she generally calm around dogs?
How does she act after dogs have a smell of her?
Do dogs try to pin her when she is down or pretend to hump her?
I guess to build confidence you get in there and protect your girl. If some males were intimidating my female when she was a pup I'd be in there intimidating them and protecting my girl.
You can also build confidence and drive through playing with a ball and string or tug.Obedience also builds confidence especially when you use it to build drive. As the dog gets into these games she can start to ignore other dogs more but you need to get in there and train her.