I've read the actual referenced paper and not the LA times piece.
This type of research is hypothesis generating. As they acknowledge, this shows a correlation, not causation. The big question here is why were these women taking paracetamol/acetaminophen? Was it the drug or the cause of their pain/fever/discomfort that is more relevant to the outcomes observed?
As part of a cohort study, this was just one outcome measured. It is not clear to me whether this was a outcome measure they had intended to analyse from the outset or whether it was post-hoc. This makes a huge difference to the credibility of the finding
With regards to the relative safety of paracetamol and ibuprofen, well, they are totally different drugs. For most healthy, young people, in short courses both will cause little harm, with paracetamol having the edge in safety. Ibuprofen causes significant, undetected kidney and gastrointestinal injury. Paracetamol can cause liver failure, but generally only in overdose, prolonged use or when associated with poor nutritional intake. A healthy person taking the odd dose will not develop liver toxicity. Prolonged administration of ibuprofen will invariably cause damage. For paracetamol it is less clear.
Source: I prescribe and manage these drugs and their side effects daily. Pain management is a core part of my day job.