"Maintain" is just a synonym for "keep."  That's the word they used in some statute, probably a century ago, to mean "living there and not moving away."  He's obviously a NYC resident for tax purposes, and for "tax home."  No indication that Dad also lives in the Manhattan apartment, but it suggests that the taxpayer may be a long-time New Yorker.  He could be staying at a different hotel in Queens every month and still maintaining his NY residence.  

The NYSCPA website has an interesting article about this definition which cites a case that is different from this one:

"In Matter of Gaied v. Tax App. Trib., 22 N.Y.3d 592 (2014), the taxpayer maintained an apartment exclusively for the use of his elderly parents. The taxpayer did not keep any personal items at the apartment, and in those instances where he did sleep there to take care of his parents’ medical needs, the taxpayer would sleep on the couch because he did not have a bed in the apartment. The Court of Appeals stated that for a taxpayer to qualify as maintaining a permanent place of abode, he must have a “residential interest” in the dwelling, and that “there must be some basis to conclude that the dwelling was utilized as the taxpayer’s residence.” The taxpayer’s relationship to the dwelling did not rise to this level.

Thus, the sheer ability to access and use a dwelling is not enough to establish a residential interest if the taxpayer does not actually do so. It, however, remains possible that a residential interest might be established by virtue of the ability to use a dwelling that is primarily utilized by a family member. Under the guidelines, determination of whether an individual’s use of a residence rises to the requisite level is based—in part—on the size of the dwelling, whether the individual’s personal items are kept there, and the extent of the individual’s use."

https://www.nysscpa.org/most-popular-content/statutory-residency-in-new-york-what-qualifies-as-a-per...