Lilypad & large number of LEDs

Good Afternoon! I am very new to working with micro-controllers!

The idea isn't new but I want to connect a Lilypad Arduino 328 Mainboard and a Pulse Sensor to animate multiple LEDs to the heart rate. I need it to run with a power supply and be portable.

My design right now has only 18 lights. I'd love to have more! To be able to fade the pulse and make it more ascetically pleasing to the eye. Here is a picture of how I am planning the LEDs with only 18 LEDs (one for each available pin evenly).

I've been reading about LEDs and have done multiple searches on running the Lilypad LEDs parallel and understand that most LEDs need 20mA. If I'm reading the Lilypad LED's data sheet right, do they run at 30mA? And each pin can only handle 40mA before damaging the pin?

The power supply only puts out up to 200mA so that means I can only have 20 LEDs total?

Any help would be greatly appreciated. I feel like my questions are scatter brained but I'm trying to wrap my brain around the relation of Voltage to Current and how the battery pack relates to the Lilypad and the LEDs separately. Any recommended resources would also be greatly appreciated!

TL;DR Can I do a project with the Lilypad Arduino 328 Mainboard with more than 20 LEDs?