Siman 1 Seif 6:

Parashat HaKorbanot should be recited during the day

• The Parashiyot of Kiyor and TerumatHaDeshan may be recited at night. (M.B. 17)
• A person who doesn’t have time may reciteParashat HaKorbanot at night. (M.B. 17)However, according to the mekubalim it isprohibited since mikrah is not permitted atnight except for Thursday night. (Birkeh Yosef 1:13)
• Parashat Hakorbanot can be recited from Alot Hashahar (dawn) (Halacha Berurah 1:16)
• The minhag in our times is to recite justParashat HaTamid and Ezehu Mekoman andnot the Parashiyot of other Korbanot.(Halacha Berurah 1:15)
• The Shelah writes that Parashat HaKorbanotmay be recited on Shabbat and Yom Tov, butone should not read the accompanying יהי רצון
  (M.B. 17).

Siman 1 Seif 7:

For all korbanot that may be offered voluntarily one should ask that the recitation should be considered as if the actual korban was offered.

• There is a disagreement whether there is a יהי רצון to be recited after those korbanot that arenot offered voluntarily. (M.B. 18) The minhagof Sefaradim is not to recite יהי רצון(Halacha Berurah 1:20)

Siman 1 Seif 8:

The pasuk of ושחט†אותו†וגו†should be recitedwith the Parashat HaKorbanot.

• Nowadays this pasuk is read following theParashat HaTamid. (M.B. 19)
• The Midrash explains that this pasuk remindsHashem of the akeda. (M.B.20)

Siman 1 Seif 9:

There are those who have the custom torecite, in order: Parashat HaKiyor,Parashat Terumat HaDeshen, Parashat HaTamid, Parashat Mizbeach HaKetoretand then Parashat HaKetoret.

Shulchan Aruch Siman 1 Seif 6 

יש נוהגין לומר ... ואחר כך פרשת התמיד There are those who are accustomed to 

recite … and afterwards [they recite] Parashat HaTamid 

The Beit Yosef writes that the reading of the parashiyot of the korbanot provides a degree of atonement. The Chikrei Lev (או״ח ס׳ א) also writes that the reading of the parashiyot of the korbanot is not the same as having actually offered a korban. When the Gemara (Menachot 110a) states, “Whoever engages in the study of the Chatat [it is considered] as if (כאילו) he offered a Chatat,” the Gemara should not be taken literally. It is only “as if” he offered a Chatat. When the Gemara uses the term כאילו it is meant for emphasis rather than to be understood literally. An example of this is the Gemara’s statement (Sotah 4a) that one that becomes angry is considered as if (כאילו) he worshiped idolatry. Obviously, no one would claim that someone who got angry should be stoned and the same is true regarding one who reads the parashiyot of the korbanot it is only as if he offered the korban, but it is not to be understood literally. The fact that reading the pesukim is only akin to offering a korban serves as a foundation for an important halacha. 

Shulchan Aruch (1:6) writes that the reading of the parashiyot of the korbanot should only be done during the day since korbanot could not be offered at night. Mishnah Berurah (1:17) writes that if a person will not have time to recite korbanot during the day they could be recited at night. One reason for leniency is that the Tur writes that it is good to read the parashiyot during the day but does not make it a necessity. Additionally, since the reading of the parashiyot is considered as if one offered the korban despite the fact that many prerequisites for a valid korban are lacking, e.g. a kohen, a Beit HaMikdash, it is also possible that the requirement that it should be done during the day could also be suspended. This is an important leniency for those people who need to pray early for it allows them to get an early start on their praying. Hacham Ovadia Yosef (Yehave Da’at 2:8) rules even though we generally hold dawn starts 72 proportional minutes before sunrise in a situation where someone has to go early to work, he can rely on those poskim who hold dawn is 90 proportional minutes before sunrise and read Parashiyot of the korbanot at that time. His son Rabbi David Yosef (Halacha Berurah 1:16) explains the reason in this specific case you can rely on those poskim since there is no beracha being recited it is just a recitation of pesukim.