Even with a Blue Badge, you can't park anywhere - parking near zebra crossings, bends, junctions etc is dangerous for other road users, no matter who is in the car. Information about applying for an using a Blue Badge here:
http://www.direct.gov.uk/en/disabledpeople/motoringandtransport/dg_4001061
As others have explained, Blue Badges are issued by local councils. High Rate Mobility is a sure way of getting a Blue Badge, but councils will look at other factors on their merits, although issuing guidelines vary between places.
It is possible to get High Rate Mobility for a child, aged 3 or over, with difficulty getting around. To get Low Rate Mobility, for guidance / supervision, a child needs to be 5 years old. Crudely because no children, regardless of disability or none, aged under 3 can walk very far, and no child under 5 has any road sense. Low Rate Mobility doesn't qualify for a Blue Badge.
An alternative way of approaching the problem of Ruth's 2 year old's hypos in the car would be to test before setting off, rather than have to find places to stop, in the same way as a driver would, although I appreciate that a child won't like extra finger pricks.