Parenting

Responsibility is easy. Just have them do their homework and it maxes out naturally.

For empathy, you can do volunteering (which is time-consuming, but available for both children and teens) and have kids play with the doctor playset.

For manners, just have them wash the dishes a lot, and set the table if you're inclined. I think teaching kids to say thanks and sorry should help.

Emotional control, you can have them write in their journal, or do activities that help regulate their emotions like playing music.

Conflict resolution is difficult IMO. There aren't a lot of activities that directly increase it. But then, I don't think it's that useful anyway, so I don't bother with it.

Responsibility is very easy let them doing their homework, schoolprojects( can buy them in build mode under children activities), teeth brushing, cleaning litter boxes, bringing out the trash, doing laundry.
Emotional Control let writing them their diary.
Empathy let them volunteer ( phone).

The only one who I would not recommend is good Manners because of a bug that let them washing invisibly dishes all the time.

Persistence. I have the children set the table, do their homework, brush their teeth before bed, write in their journals when vexed and the family volunteers. You can also have the child Sims play with the doctor kit to boost Empathy. I love that Values have been included in this GP. I have yet to experience the bug regarding good Manners.

I haven't experienced the bug with Good Manners either and actually it's a really good trait for Sims with social aspirations, as the "Polite" introduction brings them close to Friends status (better than the Cheerful introduction, I believe).

Conflict resolution is really hard to max out. When the child is in the toddler or child stage, parents can teach them to say sorry. At first, they won't want to listen, but soon, they'll always listen. After that, have parent encourage learning to say sorry.
Other way is to find random sims, get them to fight somebody and then have your child or teen try to fix the bad relationship (Friendly). Even if it doesn't succeed, it will build that character value.
Mediator trait comes useful when you want sims to stop fighting. These sims always succeed in fixing the bad relationships. These sims are also easier to talk to when sim have bad relationship with them.