There is a subtle difference between the two.
"Later Peter's daughter Elizabeth had the Grand Palace, park and
Here there is the implication that Elizabeth got someone else to expand the palace, park, and fountains for her. To "have someone do X" or "have X done" means that you ask or command another person to do X for you, rather than doing it yourself. In the past tense, you say "had someone do X" or "had X done".
"Later Peter's daughter Elizabeth had expanded the Grand Palace,
park and fountains."
"Had expanded", on the other hand, is simply the past perfect tense of the verb "expand". So this second sentence implies that Elizabeth expanded the palace, park and fountains herself - that is, that she personally performed the work to expand them.
Given that Elizabeth was royalty, the idea of her doing actually doing any sort of serious work (especially manual labor) is extremely unlikely. It's much more likely that other people did it for her, and she just took credit for it. So I would say that the first sentence is probably the better choice in this context.