Well, yes and no.
If by "better parent" you mean, can it help you to remember all the things you need to do for your kids and family, then yes.
Your planner can help you avoid double-booking events during your kids' sports practice. It can remind you to schedule doctor and dentist checkups, and vaccines to keep your kids healthy. It can help you with menu planning to feed them nutritious meals. It can remind you to take your kids to the library, to the park, and to the aquarium. It can help you schedule paying your bills on time to keep a roof over your heads and shoes on their feet.
But in the end, you still have to actually do all of these things yourself. That's what makes you a good parent.
I've definitely bought planners in the past hoping they would make me a better parent. There's loads of family- and mom-oriented planners out there in a variety of styles and formats. Some help you keep track of your busy family easily. Some help you with budgeting and money management. Some help you plan fun and memorable holidays. And yes these are very useful tools. But you have to use those tools. In the end it's just words on paper--you have to go out and DO those things you've planned.
In the past I've also bought planners that I hoped would make me a better employee, or a better student. There are plenty of planners that are structured to help you be better at a specific role, or even multiple roles (working parent, working student, etc.)
Have you ever, or do you currently use a planner specifically designed to help you in a particular role, whether parenting, at work, or while studying? Did you find it especially helpful, or not?