How parse this string into an array

If you just want the address as a single entry like myArray[1] = "Paris Ltd,5 North Street, Athens, SW1X 9LA", then try this:

<cfset myArray = [] />

<cfloop list="#theString#" delimiters="#chr(10)#" index="line">

<cfif listLen(line)>

<!--- there is content in the line, so just add the bit between " to the array --->

<cfset arrayAppend(myArray, listGetAt(line, 2, '"')) />

</cfif>

</cfloop>

If you actually want the various elements broken out, then you would need an array of structs, so that you get myArray[1].street1 = "Paris Ltd":

<cfset myArray = [] />

<cfloop list="#theString#" delimiters="#chr(10)#" index="line">

<cfif listLen(line)>

<!--- there is content in the line, so parse it out into a struct and add the struct to the array --->

<cfset addressList = listGetAt(line, 2, '"') />

<cfset addressStruct = { street1=listFirst(addressList), street2=listGetAt(addressList, 2), city=listGetAt(addressList, 3), postCode=listLast(addressList) } />

<cfset arrayAppend(myArray, addressStruct) />

</cfif>

</cfloop>

Note:  As Steve pointed out, you may need to test for listLen() on the addressList to determine whether or not there is a street2 entry in any given line.