The parenting skill is only available to Sims who are young adults and older. It can be learnt by many different methods. Like with other skills, it can be learnt on a computer using the 'Web > Read parenting forums' option. Parenting books are also available and increase the parenting skill.
Conversations and simple interactions between parents and children will not increase the Sim's parenting skill. However, most other interactions will. These include scolding and punishing children, encouraging behaviors, helping with homework and school projects, teaching toddlers skills and tucking children into bed.
As parenting is based around social interactions, being in a happy mood may help with the success of these interactions and increase the parenting skill faster.
Reading parenting books to boost the parenting skill after approximately level 7 only makes negligible increases in the skill - performing social interactions with children increases the skill much faster.
When using the "Find out what is wrong" interaction on a baby after reaching level 6 of the parenting skill, a thought bubble will appear, indicating what need they need fulfilled.
"Full Parent Mode" is activated after several interactions by a parent Sim in a row, and is indicated by a chiming noise and green stars around the Sim.