Actually, money wasn't the main reason for my parents. = NO CORRECTION

They worried about the fear of an uncertain future. = They worried about the uncertain future. 

But I managed to convince them with a detailed plan and clear goals. = NO CORRECTION

Seeing how confident I was, they finally decided to support me. = NO CORRECTION 

I could realize that persuasion is really all about making people trust me. = I realized that persuasion is really all about making people trust me. 

Based on that experience, I think I've become much better at convincing people. = NO CORRECTION

Comment: I'm glad everything turned out well for you, Dahee.