There was a ice cream parlor by the name of Reed's, Reid's or Read's. They made their ice cream cones out of tubes of ice cream wrapped in parchment
To be honest, there's not a lot of specific information out there - or there's too much and it's hard to sift through.
Historically, there are at least two defunct companies with different spellings of that name. There was a Reid's Ice Cream Company that had a factory in Blue Point, New York, on Long Island. A Google search will bring up some rather fun ghost-hunting websites, as the old factory is supposed to be haunted! Also, if you search the Library of Congress's photo archives, you will find a delivery truck from the 1920's in Washington D.C. with the Reid (no 's') company name.
Another, Reed's Ice Cream (note the different spelling) used to be an ice cream parlor in Des Moines, Iowa. Could that be it? I couldn't find much information on them online, other than a comment about their Nutty Fudge Sundae and that they used to be on Forest Drive. If this looks right, you might try contacting the Des Moines Public Library to see if they have more information.
The only current companies with that name I could find is a Reed's Dairy in Idaho Falls, Idaho and a small ice cream parlor in Henrietta, NY by the name of Read's.
Does any of this help? Without more specific information as to where you had this ice cream it's as close as I can get.