OP, I'd suggest the parents have a look at the Lorna Wing Centre, they are expert in diagnosing girls and women with ASD. They should go for the full diagnostic assessment.
www.autism.org.uk/what-we-do/diagnosticservices
It may be they haven't been for full diagnosis, just screening tests which are often much cheaper because very often they are offered by people who are not qualified to diagnose, but who can score tests to spot the possible signs, then recommend a full referral.(exception being Lorna Wing who do both)
However she told everyone that she'd read something about autism and food not touching so now she has to behave like that because she is autistic
I can see both sides of that, so will fence-sit.
If she's going to present with any autistic behaviours she reads about, life could be very interesting.
Have a look at the attached the graphic which lists the areas in which autistic people can present very differently.
The full article is here neuroclastic.com/2019/05/04/its-a-spectrum-doesnt-mean-what-you-think/
For descriptions of life with autism, lots of people recommend the Girl with the Curly Hair, I like Purple Ella on youtube.
'rebalance her equilibrium' is an unusual phrase for a 14yo to use, sounds like something from a pretentious wellness blog. Most discussions about autism and overwhelm usually say they need time to process events and importantly it can take hours or days. It's not something a few deep breaths will cancel out.
her parents say we just have to do as she says and follow her lead
Oh yeah, like that happens for all autistic teens 
No, she is a family member, a single unit among a group of single units and as such you may wish to make some reasonable adjustments for her, like allocating a quiet space for her if the events aren't to her liking when she thought they would be, but you do not need to all bend over backwards and change your whole itinerary throughout the event to accommodate her so far undiagnosed preferences to the detriment of a family occasion.
I'd give her a detailed plan of what the occasion entailed, any food choices, all activities and times and any pre-warning of noise etc. then also have a quiet space nearby like her parents' car that she could retreat to at any time if she felt overwhelmed.
I'd also warn the parents to provide backup food and drink as well as a safe space if she was likely to reject what she had previously chosen for any reason.
If she is neurodiverse, she is, she was born like that and nothing can change it, although her presentation of traits can markedly change (or not) as she matures.
Her parents seem to be hung up on autism but there are many other diagnoses of neurodiversity she could have either instead of or as well as autism.