In answer to the question that titles this post on whether parental age differences at offspring conception might matter when it comes to offspring risk of autism or schizophrenia, the answer is very possibly. As the paper by Byars & Boomsma  (open-access) reports: "Our study shows that autistic and schizophrenic disorder risks tend to be higher, in more dissimilarly aged parents, relative to parents with the most similar ages." This age disparity issue has been mentioned previously in the peer-reviewed science literature  (see here for my take).
The Byars / Boomsma paper does contain a lot more information than merely whether parents dissimilar ages might advance some risk for offspring such as adding more substance to the idea that: "Older fathers and mothers both conferred increased risk for autistic but not schizophrenic disorders" (see here for some previous discussions on this issue). They also talk about their data reflecting possible "epigenetic maternal mechanisms that control patri/matrigenic parent-of-origin effects on foetal growth or offspring brain development" alongside how age might confer more "mutational load" for sperm over eggs. For some further discussion about such issues, I would draw readers attention to an interesting post that appeared on the Spectrum website a few moons ago (see here).
A few important details to point out about the Byars / Boomsma paper: "The final sample included 1 646 092 offspring that were never diagnosed with a mental disorder during the study period while 94 177 (5.41%) were diagnosed within one of the autistic or schizophrenic disorder groups." With those kind of figures it's not surprising that this is yet another study reliant on those very useful Scandinavian registries (Denmark). As well as coding participants' parents in age brackets, researchers also included an impressive lists of co variates in their statistical modelling covering maternal 'pre-exisiting' conditions, pregnancy related variables and "education and average income". They also included a rather pretty diagram to [speculatively] convey their main message (see here).
The idea that parental age might somehow be implicated in risk of various psychiatric / behavioural labels to offspring has been around for some time. Combined with factors such as the inter-pregnancy interval (IPI), there is some quite strong population evidence emerging implicating such issues in relation to risk. I personally think this kind of work is important in light of what is being noted in population trends when it comes to starting a family for example. That being said, a lot more needs to be done on the potential mechanisms behind such trends including how sperm for example, pick up their 'deletions and duplications' over time  (thanks Robert for the link). Indeed, the word 'methylation' might also be important to this topic ...
With specific regards to the possible mechanisms behind the parental age disparity finding, I can honestly say that at the moment, I just don't have any reliable ideas as to why...
 Byars SG. & Boomsma JJ. Opposite differential risks for autism and schizophrenia based on maternal age, paternal age, and parental age differences. EMPH. 2016: 286-298.
 Sandin S. et al. Autism risk associated with parental age and with increasing difference in age between the parents. Mol Psychiatry. 2016 May;21(5):693-700.
 Molina O. et al. Sperm rates of 7q11.23, 15q11q13 and 22q11.2 deletions and duplications: a FISH approach. Hum Genet. 2011 Jan;129(1):35-44.
 Atsem S. et al. Paternal age effects on sperm FOXK1 and KCNA7 methylation and transmission into the next generation. Hum Mol Genet. 2016 Sep 23. pii: ddw328.
Byars SG, & Boomsma JJ (2016). Opposite differential risks for autism and schizophrenia based on maternal age, paternal age, and parental age differences. Evolution, medicine, and public health, 2016 (1), 286-98 PMID: 27637201