I crept up the stairs, trying not to wake my parents.
现在分词短语 trying not to wake my parents 为伴随状语,表示在“蹑手蹑脚地上楼梯”的同时“尽量不吵醒父母”。
这里的现在分词不能理解为“目的状语”。比较下面的句子,就是目的状语:
I crept up the stairs in order not to wake my parents.
类似地:
He studied hard, hoping to pass the test.(现在分词为伴随状语)
He studied hard (in order) to pass the test.(不定式为目的状语)