How does the age of a parent affect the chances of occurrence of certain genetically transmitted diseases?
In some studies, they have observed either positive or negative association between maternal or paternal age (the age of mother or father at the time of delivery) and earlier development of genetically predisposed diseases, such as diabetes type 1 and certain cancers, in offspring.

Diabetes type 1

Maternal age and diabetes in childhood (BMJ, 2010):


  ...the risk of childhood onset diabetes [type 1] increases with
  maternal age: 5% for each five years of age...the main explanations
  are that biological programming of the child is in some way affected
  by the age of the mother, or perhaps the father...


Cancer

Parental age and risk of sporadic and familial cancer in offspring: implications for germ cell mutagenesis (Epidemiology, 1999)


  We used the nationwide Swedish Family-Cancer Database to analyze the
  effect of parental age on cancer in offspring at ages 15-53
  years...Maternal age was associated with sporadic melanoma and
  leukemia, causing a 30% excess if mothers were more than 40 years vs.
  less than 20 years of age. A marginal effect of about 10% of both
  maternal and paternal age was observed for sporadic breast cancer.
  
  Paternal age increased the RR of sporadic nervous system cancer by about 15%. Accumulation of chromosomal aberrations and mutations
  during the maturation of germ cells may be a mechanism for these
  findings. In familial cancers of colon, melanoma, and thyroid, higher
  age showed an apparent protective effect, which was also noted for
  sporadic cervical cancer and melanoma.


Rheumatoid arthritis

Ages of onset suggestive of genetic anticipation in rheumatoid arthritis multicase sibships can be explained by observational bias (Rheumatology, 2007)


  There was no significant correlation between the age of RA onset and
  the maternal or paternal ages of conception.


Cardiovascular disease

In one study, they have found no significant association between maternal age and blood pressure in children. There seem to be no studies about the effect of maternal/paternal age on early-onset heart disease.

On the other hand, offsprings are at increased risk of developing a cardiovascular disease early if their parents had it early (PLoS One, 2016).



In another study, they have observed that maternal ages &lt;25 and >45 have been associated with increased frailty index: a sum of 8 conditions (cancer, lung disease, mental health problems, diabetes, heart disease, stroke, blood pressure and arthritis)

Maternal Age and Offspring Adult Health: Evidence From the Health and Retirement Study (Demography, 2012)


  In summary, net of some obvious confounders, only maternal ages below
  25 and above 45 are associated with negative offspring health
  outcomes. (see Fig 1 and Table 2)
  
  The mechanisms thought to be responsible for the young maternal
  age–offspring health link are related to the physiological immaturity
  and sociodemographic disadvantage that often accompany young
  parenthood... On the other hand, the negative association between advanced
  maternal age and adult health is thought to be driven by the
  physiological reproductive aging of the mother.


In this study, they have not checked for the age of onset of diseases in offspring, though.