... then use Inline C => Config => LIBS => '-lparser.o';

I don't think that would work - but the following config option does:

use Config;
use Inline C => Config =>
 LDDLFLAGS => $Config::Config{lddlflags} . ' /full/path/to/parser.o';
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Alternatively, once you've got 'parser.o', you could just go that one extra step and create 'libparser.a' with:

ar cru libparser.a parser.o
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Then LIBS => '-L/full/path/to/lib -lparser' should also work fine.

Cheers,
Rob