Chabad.org brings the law of not buying a house in the context of “activities that are not in the spirit of mourning”. It speaks about sheloshim for other relatives but it applies to parents for the whole 12-month period.
One should avoid activities that are not in the spirit of mourning.
For example, one may not buy a new home, nor redecorate, renovate, or
purchase new furniture, and so on, unless one will suffer great
financial loss if it is delayed past the Shloshim.
The question then is whether renting a vacation home where one will stay for a week or a few months during the year is in the spirit of mourning.
I could not find an answer to that question but did find at dinonline regarding a vacation to recuperate the following (but I could not find the sources mentioned)
See Shulchan Aruch Y:D 389:25, the Aruch Hashulchan [seif 33] explains
that only travelling with large groups which makes a pronounce festive
atmosphere should be avoided during this time. A personal trip to
recuperate is permitted.
The implication is that if there is no pronounced festive atmosphere then renting the holiday home should be allowed.