This information was initially posted by MM13 in this topic
How to get an oyajitchi.
Note - this works on all versions - V1s, V2s, V3s, V4s, and V4.5s.
To get an Oyajitchi you need to grow an Ojitchi (male oldtimer) and an Otokitchi (female oldtimer). You grow oldtimers by either refusing or ignoring the matchmaker when she offers a potential mate or just not mating with another tama. All characters will evolve into oldtimers of their sex at about 10 years of age if they have not had a baby via the matchmaker or mated with another tama of the opposite sex.
24 hours after your oldtimers have evolved they will be able to mate. Make sure you have been connecting them prior to this to get their friendship meter full. When they mate, they will both have baby boys. They stay with the baby as usual, however 1 hour after the baby has been named and looked after, instead of evolving into a toddler, it will evolve straight into an Oyajitchi, skipping the child and teen stage.
This works for V1,V2, V3, V4 and V4.5s.
Here are a few pictures from a V2 mating (click to enlarge):
An Ojitchi with his puchitchi (baby boy)
His mate, an Otokitchi with her puchitchi
The Ojitchi leaving his baby
An hour after being named, the puchitchis evolve into Oyajitchis.