Question: 1. I picked the one that was *immersed in water*.
2. I liked the one that was *painted yellow*.

Are these examples of participial phrases acting as predicate adjectives?

Answer: 
> 
> [1] *I picked the one that was immersed in water*. 
> 
> 
> [2] *I liked the one that was painted yellow*.
> 
> 
> 

Stricly speaking, they are both ambiguous between adjectival and verbal passives. 

If they are intended to convey a stative meaning rather than a dynamic one, i.e. the state of being immersed/painted was the result of some prior event, then I would say that "immersed in water" and "painted yellow" are adjective phrases serving as predicative complements in what is sometimes called an 'adjectival passive' construction. 

If on the other hand the immersing/painting describes an event (where the addition of a *by* phrase is possible), then they would be verbal passives where "immersed" and "painted" are past participle verbs.
These are predicate adjectives since the linking verb to be ("was) precedes the phrases and specifies the noun.

> 
> The trick to identifying predicate adjectives is spotting linking
>  verbs. The linking verbs include the following:
> 
> 
> ....The verb to be (in its various forms, e.g., am, are, is, was, were,
>  will be, has been, have been).
> 
> 
> ....The "sense" verbs (e.g., to feel, to look, to smell, to taste, to sound).
> 
> 
> ....The "status" verbs (e.g., to appear, to become, to continue, to grow, to seem, to turn).
>  A linking verb will always be completed by an adjective (a predicate adjective) or a noun (a predicate nominative).
> 
> 
> 

The participle phrase does describe the noun, but is an addition to the complete sentence and is not essential to complete the sentence.

> 
> A participle is a verb form that can be used as an adjective. There are two types of participles:
> 
> 
> 
> ```
> Present Participles (ending "-ing"). Here is an example of one as an adjective:
>     ...The rising tide
> Past Participles (usually ending "-ed," "-d," "-t," "-en," or "-n"). Here is an example of one as an adjective:
>     ...The risen cake
> 
> ```
> 
> Use a participle phrase to say something about your subject before
>  you've even mentioned your subject. That's cool. For example:
> 
> 
> ....Packed with vitamin C and antioxidants, oranges are a popular
>  fruit.
> 
> 
> Placed at the front of a sentence, a participle phrase is offset with
>  a comma.
> 
> 
> A participle phrase placed immediately after the noun its
>  modifying is not offset with commas (unless it's nonessential). 
> 
> 
> Put your participle phrase next to its noun. If there isn't a noun, you're
>  dangling (and that's never good).
> 
> 
> ....Having read your letter, my cat could not have fathered your kittens. [dangling modifier]
> 
> 
> ....Having read your letter, I can assure you that my cat could not have fathered your kittens.
> 
> 
> 

To answer your question of whether the two examples could be participle phrases acting as predicate adjectives, I would say no because the use is as a predicate adjective. Even though these phrases could be used as participle phrases in another construction, these words do not become a participle phrase until they are used as one.