Am I obligated to accept this parent's request to send a read receipt for their email? Would it be considered rude to decline it?
No, there is no obligation on anyone's part to send a read receipt for emails.
It may be considered rude to the email sender if they, upon encountering those who choose not to comply with the read receipt request, assume that the lack of compliance is a giant "middle finger": it certainly is possible that some portion of email senders fall into this category (but/and there are those who think it is rude, annoying, etc. to inundate email recipients with this additional hurdle, so it works both ways).
is there a certain expectation around read receipts for internal email?
Expectations are not always set in stone. In this case I suggest that, if you do not typically comply with read receipt requests, that you get into the habit early on of not doing that. Over time, people will know that "Oh, yeah, AegisCruiser never bothers with that when I send emails to them!"; some will respect that, and others will get annoyed. Having said that, if you come across some, for instance, powerful administrator at your institution, and you know that they want you to utilize the read receipt, then it's probably better to comply, or, if you cannot tolerate that, stand your ground and see what happens, possibly having in mind some other opportunities should the need arise.
Since you mentioned that you are working in the US:
As a side comment, I suggest that you do not contact the parent of your student directly, since this may violate FERPA rules: it is my understanding that, if a student has FERPA-ed their details, then we instructors are not allowed to confirm with anyone that an individual is a student in our course/enrolled at our institution. In this case, I suggest discussing with your department head how best to proceed.