Probably the easiest option avoiding manual schematic editing or programmatic .asc file hackery would be to use the command line -netlist option to convert the schematic into a netlist.
Next, script the creation of as many uniquely named 'stub' netlists as you want to run instances, each containing whatever .step or .param directives are required to get distinct results from that instance, followed by an .include directive for the previously converted netlist. As the converted netlist is terminated by a .end directive that .include must be the last thing in the 'stub' netlist.
Finally launch in parallel an instance of LTspice for each 'stub' netlist in batch mode. Results of .measure directives will be in the .log files named for each 'stub' netlist. If you need to combine waveform files from the different instances, look at Helmut Sennewald's Ltsputil.exe utility, which you should be able to get from the LTspice user group archives.