Does older production plant (with many aged or obsoleted production equipment) require bigger storage capacity for spare parts store compared to newer production plant?
I do not think there is a general rule for that. New production equipment is more reliable and parts are available more easily.
Sometimes old equipment is way more resilient and the operators can make modifications to fix issues without change in production quality. Old equipment are more robust, less sophisticated, and the parts itself are repairable. Also, old equipment is not combined on the level the new ones are, so only a part of the technology or operations affected. If it is not the bottleneck operation, you can still continue the production. In that case, you only need more parts for the bottleneck operations. In the case of a new equipment, maybe the entire process stopped since the high level of compaction and complexity of the new equipment.
So, it depends what kind of technology you assess.