There is probably a more elegant way to do that, but you can achieve it with split in a dozen lines of code. It's still string manipulation, so it's probably not what you're asking for.
Don't forget to read again how JS Dates works (the important thing is that months in JS go from 0 to 11).
Some code I made for you:
var euroStringDate = "13/05/2015 12:00:00 PM";
var splitDateTime = euroStringDate.split(' ',2); //separate date and time
var splitDate = splitDateTime[0].split("/"); //separate each digit in the date
var splitTime = splitDateTime[1].split(":"); //separate each digit in the time
var day = splitDate[0];
var month = splitDate[1] - 1; //JS counts months from 0 to 11
var year = splitDate[2];
var hour = splitTime[0];
var min = splitTime[1];
var sec = splitTime[2];
var d = new Date(year,month,day,hour,min,sec,0); //place the pieces in the correct order
And here's a jsfiddle.