While it is always necessary to replace both air conditioning components at the same time (in a split system), it’s not always necessary to replace both AC components and the furnace. The only time it really makes sense to wait to replace your furnace is if it is less than 10 years old.
It is recommended that homeowners replace their HVAC system every 10 to 15 years. Modern air conditioners are durable, but their major components will begin to deteriorate after around ten years depending on how often you use and maintain them.