Answer:
1. the night being rainy and cold → absolute phrase
2. opening the birthday card → participial phrase
3. the prizes having been awarded → absolute phrase
4. sitting beside his grandfather → participial phrase
5. having found a parking spot → participial phrase
6. the chocolate cake baked → absolute phrase
7. the war finally ending → absolute phrase
8. having attended the concert held at Stanley Park → participial phrase
9. dressed in his old uniform → participial phrase
10. the coach singing their praises → absolute phrase
Explanation:
Both absolute and participial phrases are used to modify and offer further information about something. The absolute phrase modifies an entire sentence, while the participial phrase modifies a noun. The best way to identify each is by looking at their subjects. The absolute phrase has a subject of its own, while the participial phrase has the same subject as the other clause.
Let's use two of the sentences in the question to better illustrate that:
6. The chocolate cake baked, Mrs. Patruni turned to the decorations. --> The subject of the phrase is "the chocolate cake", while the subject of the clause is "Mrs. Patruni". Thus, we have an absolute sentence.
9. Dressed in his old uniform, Dad walked in the parade. --> The subject of the clause and of the phrase is the same, "Dad". Dad walked and Dad was dressed. In this case, we have a participial phrase.