try

<style>
#MSOZoneCell_WebPartWPQ2 {display:none}
</style>

do a view source  of the page  to get the zone  id.

Here some script that will hide  the zone  that a CEWP is in unless the
page is in Design mode or a Toolpane is open:

<script>
function HideMyZone_WPQ_()
{
try
{
var InDesignMode =
document.getElementById("MSOLayout_InDesignMode");

var ToolpaneMode =
document.getElementById("MSOTlPn_View");

if(InDesignMode.value > 0 || ToolpaneMode.value > 0)
{
//Let the Zone show
}
else
{
//Hide the Zone that this web  Part is in
var webpartCell = WebPart_WPQ_.parentNode;
var webpartTable =
webpartCell.parentNode.parentNode.parentNode;
var zone =
webpartTable.parentNode.parentNode.parentNode.parentNode;
zone.style.display = "none";
}
}
catch(e)
{
window.status = "ERROR: HideMyZone_WPQ_: " + e.Message;
}
}
window.attachEvent("onload", HideMyZone_WPQ_);
</script>

I had no idea that the web  Zone was the 4th generation
parent of the web part table. I guess that's why I couldn't figure
out what to do by studying the IDs in the rendered page's HTML source.

This is great for the servers where we can't easily touch the site
templates directory.

It's too bad that Web Parts rely on HTML tables. First, they don't meet
accessibility standards. Second, the parentage of a table, tablebody,
tr, and td are not fun to work with. Divs should be sufficient but MS
didn't build it that way.