What exactly were the parks that the Buddha spent time in?
Using Google to find a Pali version of the Alagaddupama Sutta , it starts,

Evaṃ me sutaṃ: ekaṃ samayaṃ bhagavā sāvatthiyaṃ viharati jetavane anāthapiṇḍikassa ārāme.

My first guess was that ārāme is the word you're looking to translate (because it's next to the word &quot;anāthapiṇḍikassa&quot;).
Various online Pali dictionaries don't return a result when I try to look it up, but I guessed the reason for that is that ārāme is probably a (grammatical) declension of the word instead of the root.
Using this Criticial Pali Dictionary to search for ārām* (note using * &quot;Truncation sign&quot;) returns 160 words. Using the 'Next' button to scroll down until the words begin with ārām (i.e. the right accents over the vowels) finds this definition ...

2ārāma, m. [ts.], a place of pleasure i. e. a
pleasure grove, garden, or park in a monastery, or
(as pars pro toto) a monastery itself (see SBE XIII
p. 23, note 2; Geiger, Culture Â§Â§81, 175, 196);

... with a long list of references to places where the word is used.
In English a 'Park' isn't necessarily a municipal, public park: I imagine it might be a like a garden, but perhaps wilder (not all lawn and flower-beds), or a private wood. I expect it's reserved in some way (reserved for public use, or reserved for private use), not used for agriculture etc.
Similar/related/compound Pali words suggest it's used as a synonym for 'monastery', for example,

ārāma-gata, mfn., having gone to or being in a monastery
ārāma-dāna, n., giving of an ārāma

(not too surprising when monks dwell in a 'forest' instead of as 'householders').
It's also used in contexts which suggest it means delight and pleasure:

1ārāma, m. [ts.], delight, pleasure
ārāma-ramma, m(n)., the delightful, charming
or beautiful abode of a monastery

I recommend you browse the other related compound words to see other ways in which it's used; for example some of the less common words include

ārāma-(p)patta, mfn., who has reached the park
ārāma-sīla, mfn., having the habit of going to a park