What is the minimum age to enter into a hire contract without parental permission?
The usual rules of English common law relating to contracts with minors apply here.
Minors are those under 18 years of age.
Contracts with very young children will simply be void.
Contracts with older minors are not binding upon the minor unless they are contracts for "necessaries" (or in certain other situations, such as apprenticeships, that are not relevant here). On the other hand, contracts with minors will usually be binding upon the other contracting party.
What are necessaries? To quote Chitty (one of the leading English contract law textbooks):
"Such things as relate immediately to the person of the minor, as his necessary food, drink, clothing, lodging and medicine, are clearly necessaries for which he is liable. But the term is not confined to such matters only as are positively essential to the minor's personal subsistence or support; it is also employed to denote articles purchased for real use, so long as they are not merely ornamental, or are used as matters of comfort or conveneince only, and it is a relative term to be construed with reference to the minor's age and station in life."
As this quote implies, most of the case law relates to contracts for the purchase rather than hire of goods. I've had a very quick look, and can't find any cases relating to the hire of chattels (bailment). There are some cases on the lease of land, but I'd hesitate to draw any conclusions from them.
In summary, whether the contract in question is void or valid will depend upon the age of the minor in question. Whether a valid contract of hire is voidable at the option of the minor will depend upon the nature of the contract and the position of the minor. For example, a contract of hire for a medical device with a 17 year old would likely be non-voidable, whereas a contract for the hire of a video game to a 13 year old would likely be voidable at the option of the child.
Parental permission does not change the position - unless you are really contracting with the parent instead of the child. If you are contracting with an (adult) parent, then this sort of capacity question simply doesn't arise, notwithstanding that the goods may be hired for the benefit of a child.