The bee is on the Dad's hat.

sb70012I thought we just capitalize 'Dad' and 'Mom' when we are addressing them.

Correct. Capitalize and do not use a determiner (a, the, my, your, ...).

Otherwise, do not capitalize, and use a determiner, very rarely 'the', but often 'my', 'your', 'his', 'her', etc.

The bee is on Dad's hat. / The bee is on my [dad's / father's] hat. / Th

But I couldn't understand why we capitalize 'Dad'. We are not talking to dad. We are just talking about dad. So, we should capitalize dad just when we are talking to dad not talking about dad.

I'm confused here.

In #1 we are not talking to dad. We are just talking about dad. So, why is it capitalized?

Thank you.

sb70012But I couldn't understand why we capitalize 'Dad'. We are not talking to Dad. We are just talking about Dad.

There is nothing to understand. It's the name you call him when you address him. That means it's capitalized. We capitalize other names even if we are talking about them,

sb70012? I thought we just capitalize 'Dad' and 'Mom' when we are addressing them.

That is not correct. We use the initial capital letter whenever we are using the word as a name.

A: Is Dad driving you to school today, Pete?
B: Yes, he is, Mum. I don't want him to. He's a terrible driver.
A: Don't talk about Dad like that.

fivejedjon A: Is Dad driving you to school today, Pete?B: Yes, he is, Mum. I don't want him to. He's a terrible driver.A: Don't talk about Dad like that.

Thank you.
Can we also write 'Dad' with a lowercase letter in your sentences?

I mean:
A: Is dad driving you to school today, Pete?
B: Yes, he is, mum. I don't want

Not if you wish to write the words correctly. You can if you are writing about 'my dad'.

Tell dad to bring your chair in here.

Source: longman Dictionary

But why isn't dad capitalized here? It's not "your dad" too..

sb70012But why isn't dad capitalized here?

Ask the editors of the Longman Dictionary why they allow such errors to survive proofreading.