「雑用を押し付ける」ことは
ーto have some chore shoved one's way
のように表現できます。
例:
I went to school for the first time in a while, and one of the older students shoved some chores my way.
「久しぶりに学校に行ったら、先輩に雑用を押し付けられた」
New employees are often made to organize drinking parties at work by those above them.
「新入社員はよく先輩に飲み会の幹事を押し付けられる」
ご参考まで!