Technically, a checkbox may qualify as a digital signature; however, identification / authentication and storage integrity are fairly challengeable (just as a written signature on a piece of paper with a date written on it is challengeable)
Given that notarization is not required for parental consent forms, I'm not sure what sort of server security expense is justified or feasible.
How much does processing all of the paper forms cost each school? Per-student?
In terms of storing digital record of authorization, a private set of per-student OpenBadges with each OpenBadge issued by the school would be easy enough. W3C Verified Claims (and Linked Data Signatures) are the latest standards for this sort of thing.
We could evaluate our current standards for chain of custody in regards to the level of trust we place in commercial e-signature platforms.
The school could send home a sheet with a QR code and a shorturl, but that would be more expensive than running hundreds of copies of the same sheet of paper.
The school could require a parent or guardian's email address for each student in the SIS Student Information System and email unique links to prefilled forms requesting authorization(s).
Just as with e-Voting, assuring that the person who checks a checkbox or tries to scribble their signature with a mouse or touchscreen is the authorized individual may be more difficult than verifying that a given written signature is that of the parent or guardian authorized to authorize.
AFAIU, Google Forms for School can include the logged-in user's username; but parents don't have school domain accounts with Google Apps for Education or Google Classroom.
How would the solution integrate with schools' existing SIS (Student Information Systems)? Upload a CSV of (student, {student info}, {guardian email (s)})? This is private information that deserves security, which costs money.
Which users can log-in for the school and/or district to check the state of the permission / authorization requests and PII personally-identifiable information.
While cryptographic signatures may be overkill as a substitute for permission slips, FWIW, a timestamp within a cryptographically-signed document only indicates what the local clock was set to at the time. Blockchains have relatively indisputable timestamps ("certainly no later than the time that the tx made it into a block"), but blockchains don't solve for proving the key-person relation at a given point in time.
And also, my parent or guardian said you can take me on field trips if you want.
https://backpack.openbadges.org/