Verifying value before parsing
I would put the try/catch over as little code as possible.  This also makes it easier to add an else case to the if statement if necessary.  For example:

String parsedValue = DEFAULT_VALUE;
if (myValue != null) {
  try {
    parsedValue = parse(myValue);
  } catch (ParseException e) {
    log(e);
  }
} else {
  log(new Exception("Empty myValue!"));
}


This allows for more flexibility than the other version.  

As a general rule, it's best to put as little code inside a try block as possible.  Only things that can throw an exception that should be caught by the catch and connecting code.  One reason is that other code than you originally intended may throw the exception and the solution in the catch block may not apply to the new case.