这里面包含三个独立的句子，并不是一个复杂的主从句。

1. Having noticed my interest in stories （现在分词短语作状语）, my parents （主）lost（动） no time （宾）in initiating me into reading.（状） 

2. They bought many storybooks with illustrations（第一并列分句）, and whenever free,（第二并列分句的时间从句） they would read these stories to me over and over again. （第二并列分句）

3. By and by （状）I （主）had （动）a vocabulary （宾）large enough to read on my own.（定）