I have the "Snap" option on the Double Line command turned on, but it does not snap to existing points. The snap size is set at the default, 3.
LT version?
By chance are you trying to combine this with a running OSNAP setting? DLINE command turns off running osnaps, leaving SNAP (the Dline subset) to behave more like a slightly smarter 'nearest' osnap (3 for example is how many pixels on your monitor AutoCAD will use to judge you are near another object).
The OSNAPS are turned off during the DLINE command by design. A workaround for this is to activate the Object Snap tool bar via the CUI (Customize User Interface) and to click on the appropriate snap for your line.
This sounds that it would be valuable to have in the program. If you would take the time to go here:
http://usa.autodesk.com/adsk/servlet/index?siteID=123112&id=1109794
This is a feedback page where customers can make suggestions. This page is monitored by product managers who affect real change and this request can go a long way in changing the product for the better.
Regards
It takes about 1-minute to create DLINE as a menu selection anywhere you want it using CUI command: help yourself, HELP in LT is a good place to start learning how to use CUI and there are a plethora of Youtube videos and blogs on the topic if you need more help.