Since your parents don't have any other pets I would take her. You'll never know how she'll do unless you try. If she does OK you will probably find that it gets easier the more often you take her.
I definitely would not take her out of the carrier during a couple hour trip...that will likely stress her more and provide the opportunity for escape.
My first cat Onyx was an only cat and she went all kinds of places with me. We'd go to my parents house about twice a month (an hour drive each way). To NJ a couple times a year. And then to Florida when my parents moved.
She would remember each place she went and when she got there she'd go looking for her litterbox and food dish in the places they were the last time...and meow at me until I set them up. She might not eat or use the box right away, but once she knew they were in place she was happy and would go find a comfy spot to hang out.
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Holly, Misty & their friend, Jake, the dog.
Onyx, Callie May, Maggie & Kobi forever in my heart.
Yes, I know I need a new signature pic...someday...