You are assuming two students, attending decades apart would have to abide by the same rules.
It you talk to a chichéd "old person" long enough they no doubt tell you about the time their friends and they walked to <insert major city> right down the middle of the road and they never got run over or stabbed or any of that other nastiness you read in the newspaper. They'll explain this away purposefully: times were different back then.
If yesteryear's children acted out, they were beaten, sent down the mines, sent to borstal, enlisted, etc. Indeed the attitudes of Filch and Moody would suggest that corporal punishment (hanging from their wrists, transfiguration) was acceptable... at least at some point in the past.
That is all to say boundary rules were less common in Riddle's era. Children had more ultimate responsibility for their own whereabouts, behaviour and the ramifications of getting either wrong.
Indeed the very notion of a "parental consent form" or "permission slip" is of a much more recent time. It wasn't until 1974 that many of these things were legislated (or suggested) by UK muggle law... Around the time when aforementioned "old people" might be telling you that 'elf 'n' safey [had] gone maaad.
Moreover there's certainly nothing canon to uphold you base assumption. Nothing suggests this permission slip rule was an old or immutable law. Given that many of the protections the students had at Hogwarts were of Dumbledore's implementation, and that he wasn't even the headmaster when Riddle was attending, it seems most likely that this rule may have been implemented by him as it came into fashion in the muggle world.
But if the rule had existed, I do not think Mrs Cole would have been his guardian, at least not for this. I went to a school that had long-term boarders —the sort who only went home once or twice a year— and they were essentially signed over to their house -masters and -mistresses. They were the guardians. They would be the people who make guardian and medical decisions in place of the actual parents.
It might sound weird but it actually makes a lot of sense. If you have a child in your school with parents who could very well be in another country, you need the executive power to make decisions like this.
But anyway, following this logic Tom Riddle would have needed the signature of Horace Slughorn (or whomever the Head of Slytherin was in Riddle's time).