The study you cite found an rate of autism of 6.3% among circumcised boys, and 1.5% among uncircumcised Danish boys. Another study also examined Danish boys, and got different results:
In our patient population, we found that 7.2% (95% CI 3.4–11.0) were
diagnosed with autism spectrum disorder (ASD) as a concomitant disease
(Table 3). A recent Danish study calculated the prevalence of ASD in
Denmark to be 1.5% among boys, but a higher prevalence of ASD in
other background population studies has recently been calculated to be
in accordance with our findings.
In other words, 1.5% background is an abnormally low result.
A PubMed comment on the original study also points out that it misinterprets the mechanism that was being examined by one of its sources.
The finding in this cohort of 342,877 boys born between 1994 and
2003, that circumcised boys were more likely than intact boys to
develop ASD before age 10 ... concurs with
our study findings. It is, however, important to note
that in our study we did not focus on the psychological consequences
of the circumcision procedure as the causal ASD mechanism, rather we
were using circumcision exposure as a proxy for the potential
paracetamol (APAP, acetaminophen) exposure that may occur with the
procedure. Our study aim was to explore at a population level the
hypothesis of a relationship between paracetamol exposure and ASD.
A perplexing finding in this Frisch and Simonsen Danish study is a
statistically significant association between circumcision and the
development of autism in the cohort of boys ages 0-4 (hazard ratio
(HR)= 1.80, 95% CI 1.25-2.60) while only a very weak association for
the boys ages 5-9 (HR=1.15, 95% CI 0.75-1.77). This finding can be
explained by the paracetamol hypothesis. The boys that were 5-9 years
of age would have been born in 1999 or prior and, based on the timing
of guideline development, would likely not have been exposed to
paracetamol with the procedure, while the younger boys would likely
have received analgesia.