Reading divrei Torah (to which we will limit this answer) during davening raises several issues: 1) hefsek – an improper interruption in the midst of performing a mitzva or reciting a text; 2) lack of concentration on the matter that one is involved with; 3) creating a negative atmosphere or precedent, which may be improperly imitated; 4) derech eretz. Let’s now analyze each issue.

1) Interruption – One may not speak even divrei Torah in the following places: a) P’sukei D’zimra;[1] b) Kri’at Shema and its berachot;[2] c) Shemoneh Esrei.[3] The rule is that “hirhur lav k’dibur dami” (thinking is not like speaking).[4] Therefore, according to most poskim, thinking and even reading, while often inappropriate when one should be concentrating on tefilla, are not considered formal interruptions.[5]
2,3) Lack of concentration, atmosphere – In the midst of kri’at haTorah,[6] Kaddish, Kedusha, and Shemoneh Esrei, one should not be thinking about other matters. Regarding chazarat hashatz (repetition of Shemoneh Esrei), the Magen Avraham[7] cites two opinions regarding whether one who will be careful to answer “amen” properly is permitted to learn. The Mishna Berura[8] prefers the strict opinion, out of concern that others will learn to abuse the leniency. The Igrot Moshe[9] points out that at least nine people must be listening to every word in order to entertain relying upon the lenient option. One is permitted to learn between aliyot, if he is careful to stop as the next aliya begins.[10] Before the beginning of chazarat hashatz, most poskim permit silent learning.
4) Since derech eretz kadma la’torah (respectful behavior precedes Torah study), it is improper to read during the d’var Torah of the rabbi or a fellow congregant.

Although parasha sheets are sometimes read when it is forbidden or questionable to do so, they seem to have replaced a lot of talking and mundane thoughts in many of our shuls. Therefore, on the whole, they seem to have added more to the spirituality of our batei knesset than they have detracted.