ATC need to have clear airspace 1Nm around the drop zone for a drop height less than 10,000ft and then it increases above that (i think its 1.5NM for FL drop heights)
We operate in and around chutes all the time. Chutes on one side of the strip and aircraft on the other. It would not be uncommon for drop clearance to be given with an aircraft flying at less than 1 mile so if they are being pedantic about giving clearances that is a little average / poor form as it works well / safely elsewhere.