Entitlement to parental leave allowance when changing jobs?

There is no qualifying period, so you are entitled to the parental allowance from the first day of employment, although there may be local agreements at the university where you will start, so it is best to check what applies there. The basic rule is that you can receive the allowance for a maximum of 360 days per child and you must request the allowance no later than 2 years after the month to which the leave relates.
Regarding the age of the child, the Parental Leave Act applies, i.e. as long as you are entitled to parental benefit from the Swedish Social Insurance Agency, you are entitled to the supplement. I also throw in a disclaimer here that there may be local agreements that say otherwise.