In some of the example phrases of my textbook, these two sentences appeared.
弟はいつもコンピューター_?_ばかりしている。 Here where I used the bold question mark I'd usually expect the particle を to indicate the direct object. But it's not there. Does this happen regularly? And if so, under what circumstances is it allowed?
最近忙しくて、テレビのドラマとか映画とか見る時間がありません。 Same as in 1, just in a different construction so that you can explain the different reasons for the omission of を, if there are different reasons at all.
Particle omission (助詞落ち) depends not only on what it follows/it is followed by, but also on the "environment" (what particle is used in the sentence etc.)
First, you have to know that in "old japanese", combining most particles (if not all) was the correct way, and you can still encounter some of them in modern japanese, like をも, をは (をば) etc. Not mentioning all the double particles that are used all the time like からも, までも etc.
Anyway that being said, in modern japanese some double particles like をも have been contracted in も with を being dropped (even though you might encounter をも written somewhere once in a while). And ばかり acting as a particle, you will often see/hear ばかり used alone with the を dropped. It is the same case with だけ, くらい, ほど etc. which act as particles, so it is very common to use them by themselves and in some cases more natural. The same goes for とか. But remember that this is not the case all the time, 「とかを」, 「ばかりを」, 「をばかり」, 「だけを」, 「だけが」, 「とかが」etc. are still used even though some of them are (a lot) less common (thus unnatural).
Secondly, particles can be omitted (or changed but this is another subject) depending on the other particles and words used elsewhere in the sentence, or just depending on the sentence itself (politeness, if it's long or not etc.). But most of the time this is in casual speech, hence not correct grammatically. Example:
映画みたよ X (even though it is widely used)
映画を見たよ O
