There is a section of the community who are sensitive to some of the chemical preservatives - I've seen first hand the contact dermatitis in a friend of mine (that sounds so mild ... their skin would peel off) if they used a particular set of preservatives (the formaldehyde donors). And, as like attracts like, they knew other people with the same reactions. When the parabens were touted as endocrine disruptors (and bans of these two types of preservative systems was sought for baby lotions in some countries), there was a serious push towards a chemical free preservative system.
(In some quarters, the complexities of preservative systems meant that all synthetic preservatives went into the same bucket)
This is recognized as a serious problem - the dangers of removing preservative systems altogether (from cosmetics) would be horrendous!
I personally witnessed to a commercial product with a preservative failure (just recently, in a product manufactured by a large overseas company) ... the product was sold as an antibacterial spray for wound healing. The person who purchased the spray used it dutifully, but kept getting inflammation and infections in the wound. When they increased the use of the spray to try and stop the infection, it got worse, not better. By this time just over a week had passed. They decided to check the spray bottle, and took the lid off. Inside the bottle were ropes of black slime ... the spray did not contain a mould inhibitor
So I understand both sides - some preservatives will cause a reaction in some people, but not preserving is simply not a viable choice.
Fortunately, our understanding of biological systems is always increasing, and now (with the ban of formaldehyde preservatives in some countries, such as Japan and Sweden), the ban on some paraben preservatives in products in Europe, and more extensive bans of paraben preservatives in products for babies and young children in at least one country - Denmark), there is a real push towards finding mild, safe, yet effective, preservative systems. I personally think this isn't a bad thing, on the proviso we don't discard the idea of preservatives altogether.
For anyone interested in having a read, these two articles might be worth a look:
From the Australian Society of Cosmetic Chemists (fairly balanced):
https://ascc.com.au/preservatives-used-in-personal-care-products-2/
From Chemical & Engineering News (heavily biased):
https://pubs.acs.org/cen/coverstory/88/8820cover.html
On the preservative that is listed as Parfum, or Fragrance, I think that one is Naticide (or similar), which is cannot be listed by products trade name, and has an INCI of either Perfume or Fragrance
https://www.sinerga.it/en/raw-materials/products/microbial-inhibitor/naticide
In Europe (and I think here in Australia, although I would have to check) the claim "preservative free" cannot be made if a product contains Naticide.
The group of people I referred to at the top of my post also tend to avoid products containing the catch-all/generic "Fragrance".