Is it illegal for a 13 or 14 year old to have a sex free relationship with an 18 year old?

What would an 18 year old have in common with a 13 year old?

Not really since almost all 18 years will want to have sex. So the issue is not what they say, but what will really happen a few months into dating.

What about when they are 19 and 14 or 20 and 15.

Sorry sex will enter the picture long before the child is old enough to have sex.

Next the parents of the 13 year old has to agree for the dating, I know of no good parent that would even consider it happening

It isn't illegal, but any adult can cause huge trouble for the 18 year old (who is adult in the eyes of the law). The charge can be endangerment for any number of reasons, such as going to a party where cops are called, or getting stopped for speeding, or even just sneaking out past the orders of a parent. The charge can be statutory rape, even if there was no sex, if someone thinks there was. It's a situation where 'innocent until proven guilty' flies out the window. It's nothing but trouble.