What sort of sex-determination system would be required for males of a species to only result from parthenogenesis, whereas females only result from fertilization? I am not aware of a system exactly like this on Earth.
After wikipedia: Arrhenotoky (from Greek -τόκος -tókos "birth of -" + ἄρρην árrhēn "male person"), also known as arrhenotokous parthenogenesis, is a form of parthenogenesis in which unfertilized eggs develop into males.
a/α determination system
This sex-determination system is wholly made up. Under this system, there are two types of allosomes: a and α. One sex is heterozygous (a/α), the other sex is homozygous (a/a or α/α).
Unlike other sex-determination systems, but like mating types in algae and fungi, gametes carrying the same allosome cannot fuse due to special incompatibility proteins on the cell surface. Parthenogenesis bypasses incompatibility through chromosome replication within an oocyte, a form of automixis. This will produce a homozygous zygote.
The heterozygous sex could only be produced through sexual reproduction while the homozygous sex could only be produced through parthenogenesis.
If the homozygous sex is male and the heterozygous sex is female, then this provides the circumstances desired by the original question. Sons will have only mothers, while daughters will have mothers and fathers, but grandsons may have maternal grandfathers.
If the homozygous sex is female and the heterozygous sex is male,
then this results in an evolutionary dead end because the female
population will effectively be clones of one another. Daughters will only ever have mothers, never fathers nor grandfathers.