If I work for a subsidiary company, should I say that I work for the parent company on my LinkedIn profile?
I think it depends largely upon how the company is branded. If the corporate website, logos, etc. of the company you work directly for is branded with "ScissorCo, a division of MegaSlice Cutlery" then I would use that on LinkedIn.
If the primary branding is just ScissorCo, and the "we're a subsidiary of MegaSlice Cutlery" is on the "About Us" page only, list it as ScissorCo.
Look at the LinkedIn profiles of your colleagues as well, see what they've done.
On Linkedin, search for the name for the subsidiary. Then search for the name of the parent company. Read several profiles at random. See for yourself how the authors of these profiles handle the subsidiary/parent company relationship. There is most likely a de facto consensus there and you should be able to identify it. Then go with the flow and do whatever everybody else who works for the subsidiary is doing to represent their employment at the subsidiary on Linkedin.