Does http::response_parser inherits limits from source parser?

Yes. The limits are part of basicparser which defines only defaulted move and assignment.

It's also possible to test, using the friend-ness of basicparsertest:

#include 
namespace boost::beast::http {
    class basicparsertest {
      public:
        static void foo() {
            auto parser = std::makeshared();
            parser->headerlimit(std::numericlimits::max());
            parser->bodylimit(std::numericlimits::max());
            auto newparser = std::makeshared(std::move(*parser));
            assert(newparser->bodylimit == std::numericlimits::max());
            assert(newparser->headerlimit == std::numericlimits::max());
        }
    };
} // namespace boost::beast::http
int main() {
    boost::beast::http::basicparsertest::foo();
}

All asserts pass.