How to stop the reminders icon from flashing?

You need to fix what is causing the reminders. 

For example, for the parts, you can double click the reminder and uncheck this box : 


Alternatively, instead of doing it one by one, you can go to the "Parts inventory" section, select all of the parts, right click the grid and use this menu to set all the parts reminders inactive:


For the "Service due" reminders, just double click it and change the "Next due" values. In the following example, it's the meter that is triggering the reminder (the next due date is not reached yet) :