Subscriptions do support FastSpring's discount mechanism like any other product. By default, the discount will apply to the whole lifespan of a Subscription.
Create a discount
Go to "Offers" and choose "Create Discount" from the menu and create a discount. Usually a discount will be granted in conjunction with a coupon. Thus, after creating the discount simply add a "Coupon Condition". If you want to restrict that discount to only one product, go ahead and add a "Product Condition".
Test your discount coupon
Open "Store Testing", switch to the "Optional Parameters" tab and enter your coupon code. Go back to "Testing Links" and open your product.
Restrict discount to first or regular period
FastSpring lets you restrict a discount to the first or regular period. Simply open your discount and add a new "Order Environment Condition". This condition allows you to restrict your discount to a number of different order environment values. One of them is the subscription period. Choose either "First Period" or "Regular Period" here.
Apply discount coupon to an existing subscription
As already mentioned, by default the discount will apply to the whole lifespan of a subscription. You can add conditions like discount range, period restriction, coupon code, etc. but they do only define under which conditions a discount will be granted. If you like to define how long a discount lasts you can define this by editing a subscription which has been created via an order (e.g. via recent orders). There’s a duration field below the coupon field which defines the (remaining) number of periods a discount lasts.