Many parasites co-evolved with their hosts to evade immune detection, for instance hookworms have an outer layer that tricks immunoglobulin (antibody) response.

If the immune system doesn't detect a foreign body, then it won't mount a response.

Your cells are constantly presenting antigens on their surface (look up major histocompatibility if you want a more in-depth explanation of how cells express themselves to the immune system).

Unfortunately, the antigen expression of a donor is rarely 100% identical to the recipient (exceptions would be transplants made from host tissue, or from an identical twin). There are synthetically biocompatible materials, though.

If the antigen expression doesn't match self expression, then there's a good chance that an immune response will occur (look up AutoImmuneREgulator, AIRE gene and thymus function for more details on how autoimmune regulation occurs, along with "immune sensitization" and "immune silencing").

The little differences of the proteins between tissues of different genotypes (e.g. the genetic difference between donor and recipient) that get chopped up and presented on the outside of a cell for the immune system to recognize are enough to cause a transplant to be rejected by the body.

There are some immune privileged sites, though, such as the eyes. Which is why, to my knowledge, cornea transplants are rarely rejected by the host immune system.

Parasites often try to make themselves immune privileged, so to speak. A number of parasites will secrete mucus containing host antigens, giving themselves a bit of a slimy wall of protection. That's one strategy which I remember, but I'm no parasitologist.