If someone skips parts of shacharit in order to say shemoneh esrei with the congregation, (when) should the skipped paragraphs be made up?
First of all, the berachos upon pesukei dezimra (Yishtabach and Baruch Sheamar) cannot be made after one has said the Amidah (Shulchan Aruch 52). The question, then, is about the pesukim/mizmorim.
The Shulchan Aruch there writes that one may, after finishing davening, go back to say the parts of pesukei dezimra that he skipped, and it sounds a bit like he feels that this is the proper thing to do. This also seems to be the position of the Mishna Berurah (52:9). The Aruch Hashulchan (52:4), though, believes that according to the Shulchan Aruch this is merely optional. There doesn't seem to be a time limit, as the Shulchan Aruch's reasoning is that saying pesukei dezimra is no different than saying tehillim, which can be said at any time.
However, the Aruch Hashulchan himself (52:5) believes that one is prohibited from saying pesukei dezimra after prayer, and he insists that this is the position of the Tur and geonim. His reasoning is that "pesukei dezimra was only established for pre-prayer" (in addition to kabalistic reasons). Another reason may also be that it's improper to praise God after praying to Him (see Maharal in Nesiv Ha'Avodah ch. 12 based on Megillah daf 18a)
In shulchan aruch siman nun beis it says "afterwards you should say the psukei dizimra without the brocho before and brochoh after (meaning boruch sheomar and yishtabach) - (see shulchan aruch harav there where he explains why you don't say the brochos -) there are two opinions whether you should say it after davening or not therefore the halachah is to say it without the brocohos see also mishnah brurah ois tes where he uses the words ואחר התפלה and after davening...
So you say it after you finish davening.
But see mishnah brurah and others on this who say that there is a custom to not skip any part of davening even for the benefit of tifilah bitzibur.
See also igros kodesh of the lubavitcer rebbe cheilek yud ches page pay aleph who writes that there are those who are against skipping davening even to daven shmonah esrei with the minyan (even if you came on time and the minyan is davening too fast for you).