You don't need to upgrade your account if you mean buying a subscription.  I think you're just out of land/stall space.  If you go here https://horsefable.com/estate/color-time you can see that you've only got a 10 acre estate and are maxed out on what you have on your land.  It also shows you only have 2 stalls right now, so you can't board more horses at home without buying more land and building more stalls which you'd do here https://horsefable.com/construction by buying more land first then a building gives you more stalls.  You should be able to do outside boarding though, not sure why you can't do that.