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The NYT says no. But let me ask you Spotify; I'm asking to give you more money. Do you want it?
Unfortunately not. You can't apply additional discounts to a Premium for Family plan.
If you'd like to switch from your Premium subscription via NYT to Spotify's Premium for Family plan, then you'll have to cancel your Spotify Premium subscription via NYT first.
After your subscription has expired and you're back on the Free plan, you can upgrade to Premium for Family at https://spotify.com/family.