To preface this, because parli competitors from different states don't mix all that much before PITOC, the three main parli states (California, Texas, and Colorado) have independently developed different perceptions of parli, different implied rules and theory, etcetera. In fact, there were issues at PITOC with teams from different states holding each other to conflicting standards.
Since I do parli in CO, I consider that evidence should never be cited. The only citations I'm okay with are when you provide a fairly common quote. Colorado also seems to tend to avoid the standard debate formats more than other states, and has embraced a more relaxed atmosphere during parli rounds. This is not the case in other states, and so my advice should be viewed with that in mind.
Is there anything that is common in TP that we should avoid in parli? Evidence presses will get you nowhere in parli, since you can't cite anything. Parli is generally going to be less fact based and more value based, even on the policy resolutions, so don't make the mistake of thinking that your numbers will win you the round. Just subtly concede a fact (not necessarily the point, though) if you realize that the other team is probably right.
How do you use points of information effectively? POIs are rough, because you have to find a way to fit your entire TP chain of questions into one question. The best rules for POIs are these: only ask questions with obvious answers (the judge will automatically think of the response that you want), and only ask questions that are key to building your case. Clarifying questions will hurt you a lot more in parli than in TP, in my experience.
What are ways to format gov first speeches that you find to work well? The standard format, at least in CO, seems to be definitions - weighing mechanism - plan - justifications/contentions for policy rezes, and the same format minus the plan for fact and value resolutions. I tend to like that format because it presents your intentions right away, making your case memorable and therefore clear, and because it makes it harder for the opp to target your weighing mechanism and plan with POIs, since that's in or just out of protected time.
How should we use the 15 mins of prep time Stoa rules give? If you prep as a team, one of you should be on Google while the other writes down ideas and formats them into either a case or DAs, depending on whether it's a value or fact rez or a policy rez. Both of you should be brainstorming out loud. If you prep with friends or a club, the same strategy applies; one or two people on the internet, the others writing.
How do you warrant your arguments? Ideally, you want to stick with using facts that everyone knows. However, when that doesn't work out, my approach is to use only arguments that I can prove logically by using other more well-known facts as a foundation. This usually takes the form of a variety of historical precedent argument.
Does TP theory apply at all (like DA structure, T, and stock issues) or should we avoid that? TP theory works great, you'll just want to stay away from the technical terms at all costs, because chances are fairly good that the other team won't understand them even if the judge does. You probably already know that, but as a good portion of parli teams are LDers, it bears throwing out there. LDers use similar theory to policy, but they use different terminology.
The biggest difference is going to be the use of weighing mechanisms. Whether you're gov or opp, in parli you don't want to go in without a WM that you can defend.
What do you find to work well in rebuttals? Don't worry about dropping a few things in the rebuttals. That's not quite as big an issue in parli as in other forms of debate simply because both teams are almost guaranteed to bring up at least one argument that really has zero relevance. It's a side effect of the short prep time and the sometimes bizarre last-minute advice from coaches and fellow competitors. The only thing you want to worry about in the rebuttals is leaving the judge with a clear picture of the most important issues viewed through your WM. It's better to emphasize your WM/value than to "go down the flow," although that's important, too. But start by focusing on your WM and your reasons to prefer it to the other teams' WM, and make sure that you fit your voting issues under it.
My guideline is, if you ever have to ask yourself if you're doing something creepy, you probably are. But then go ahead and do it anyway because being creepy is fun.