you dodged a bullet, glad your girl seems perfectly ok, maybe the grass under her head cushioned the impact of your knee
you didn't mention: at the park do you use an enclosure (several grids depending on how many you have out) and make sure they have shade? and a water source or fresh (wet) veggies?
I don't want to "pile on" but your pets rely completely on you to be as knowledgeable as possible ~ do you check the greens before exposing them? check all the sources you can, the plants growing wild are known as "forages"
sounds like your Dad is a natural ally for you, be careful not to alienate your Mom, she may just need a bit of education to see that Guinea Pigs aren't the usual "rodent", and in time she may gradually come around... that's what happened with MY Mum
If they see you online researching here and at Guinealynx, and plenty of other sources, they will know you are serious about the best care possible, and maybe that will impress them to "give in" a bit more
Find out in advance what the Vet visit will cost. Sadly, they are very expensive even just for a wellness exam. Baseline bloodwork gets recommended, typically doubling the amount. Devoting yourself to Guinea Pigs is an investment in both time & money, maybe your Dad would like to "co-care" with you?
It was brave of you to post this accident here, you obviously love your guineas as you got them a bigger cage, I hope everything works out well for you