You are confusing two different types of phrase: relative clauses (finite adjuncts to the noun) and participial phrases (non-finite adjuncts to the noun).
As regards possible ellipses of the former, the relative pronoun can be omitted where it stands as either object or complement within its clause, e.g.
 This is the book I was reading.
 This is the book that I was reading.
 I am not the person I once was.
 I am not the person that I once was.
but not where it stands as subject. Hence we cannot have e.g.
 *This is the car was parked outside her house.
 This is the car that was parked outside her house.
Participial phrases, contrary to popular belief, are not simply abbreviated relative clauses but a distinct category of postmodifier, as is easily demonstrated by the fact that e.g.
 Anyone belonging to this organization will be arrested.
does not represent an ellipsis of
 *Anyone that is belonging to this organization will be arrested.
since, the verb 'belong' belong stative, no such sentence is possible!