How do I know which alleles the parents have?
Let's go step my step here.

Going by your notation I'll call the square as Male and Circle as female.
So from this information I can confidently say that you disease is not X linked dominant. 

Why?

Because Females carry XX, that would mean to be X linked dominant she had to be X'X' for the disease allele to manifest it's phenotype. But if it were X'X, it would lead to a loss in fitness. And since the Male is normal and XY he does not carry the disease which tells us that the disease is definitely not X linked dominant.

Therefore in a cross of X'X' and XY you get individuals X'X and X'Y in the ratio 1:1

Coming to AD Autosomal Dominant may mean A'A' as the disease allele is present in homozygous condition. A heterozygous condition would lead to loss in fitness, not necessarily a disease.

A Father may be AA or AA' 

so this cross would give you individuals 


AA' for A'A' x AA
A'A A'A' for AA' x A'A'


so in this case you always get the disease in autosomal dominant, what remains is recessive.

AA or AA' Father x A'A' Mother

Products AA' for AA x A'A'
or AA' and A'A' for AA' x A'A'

which tells us that the disease is actually Autosomal recessive on the mother.

The father's genotype for the allele is AA or homozygous dominant

and the disease is autosomal recessive for the disease in the mother.