The contrastive は can mark the scope of negation.
(From the context, the は in ジャックは is the topic/thematic particle.)
Compare:
ジャックは多くのパーティに行きます。-- 肯定文/affirmative sentence
ジャックは今日はパーティに行きません。-- 今日 is negated. (implying he might go on another day)
ジャックは車ではパーティに行きません。-- 車で is negated. (implying he might go by some other vehicle)
ジャックはパーティには行きません。-- パーティに is negated. (implying he might go to some other place)
As you can see, は marks/highlights the negated element in a sentence.
Back to your example...
ジャックはあまり多くのパーティにはいきません。
Here, あまり*多くのパーティに ("to so many parties") part is highlighted and negated. It implies Jack goes to parties, but doesn't go to that many parties.
Regarding this usage of は, these threads might help:
*あまり as an adverb is followed by a negative word (here ない), meaning "not so~~, not very~~, not really~~". For more on this, see this post on Negative Polarity Items.