If there are any valid parallels, I'm not sure what they are.
Sorry about the delay in responding - I was out of town and wanted to give this some careful consideration.
So to preface this, I'll say that I believe there are meaningful parallels - but of course there are quite a few differences, as well. I'm not arguing that we are Rome, but that there are sufficient similarities that we can usefully learn a great number of lessons from their mistakes.
Let's begin with situational parallels. The most obvious one is that we are both running large, world-spanning empires. There's an obvious difference in that most of American dominion is not being enforced by invasion and maintenance of garrisons; we expanded first as an economic sphere of influence, and have used military action rather judiciously to underscore our authority rather than establish it.
Nonetheless, we are in the position where a significant number of countries have their controlling legal authorities largely or wholly dependent on American support for their continued existence, and even more countries which are economically entirely dependent on us. Moreover, we've now started to assume direct political control in multiple foreign theaters - Iraq and Afghanistan being the most obvious cases. I believe this is a strong situational parallel because it requires American policy-makers to (directly and indirectly) determine day-to-day policy in distant areas.
I think this sort of thought can be taken fairly far, to find various technical parallels which simply stem from the requirements of imperial administration and economics.
But the differences may be more interesting than the parallels. We have three significant advantages over Rome, which I could describe as learning from their mistakes. First, we have an ordered mechanism of succession, and in fact an anchoring of the government in a "basic law" to which the people are actually loyal above their loyalty to individuals. One person takes office, another leaves it, and this occurs in an ordered and predictable way; I can't imagine any president having the nerve to try to sieze power beyond the limits of election, and if one did I can't imagine him living out the week.
Second, we have a systematic system of appropriations, funding and accounting. This is not to be discounted lightly: one of Rome's biggest problems was that taxation was done locally and by demand, taking local resources to fund local military expeditions and so on. This made long-range planning virtually impossible, and made responding to changing circumstances similarly difficult. On top of this, sporadic taxation tended to be crushing, and almost always led to local revolts - hardly a recipe for success. (Had Ammianus, the chief accountant to several emperors in the 4th century, lived longer or had an equally competent successor, things could have turned out quite differently...)
Third, we have access to fast, reliable communications. This means in particular that local administrators, generals and other officials are in steady contact with the center. Apart from obvious practical advantages, this makes one political difference - the situation of military units becoming more loyal to their general than to the empire, and generals starting to act in their own interest rather than those of the empire, doesn't tend to arise.
But there are some less-good differences: the United States does not have a large, established cadre of foreign administrators. Those we have tend to be hampered by policies of installing local rulers hand-picked more for their malleability than for their local connections and ability to run things. I'll take Chalabi as an example of this sort of problem; his willingness to work with the Americans is unquestionable, but he has almost no friends in Iraq, and the only way he's likely to ever get power is if he's forcibly installed. Moreover, he doesn't have systematic local ties, nor a feel for the current "pulse of the area." One of the most serious lacks in present foreign adminstration is that in far too many regions, we have occasional agents, but we don't have large-scale ties (say, with permanent employees of our government's local administrations) with enough of the local populace to keep a feeling on the local pulse. This means we're getting insufficient intelligence about what's going on in the area. Rome avoided this by having long-term provincial governors who got to really know their areas, and because the relative isolation of provincial posts encouraged intermarriage and other long-term interactions of the entire local Roman staffs with the population.
And finally, I think that there's one very unfortunate parallel, but one that's perhaps more easily rectifiable than all of the above.
There was the legend of Roma Æterna, that no external force would ever be able to really make a dent in Rome's empire, that external forces could always be dealt with by force and the might of the empire. This worked up until approximately the end of the 3rd century; but when Rome reached some functional limits on expansion, and couldn't keep going out and conquering everyone they met, things changed, and Late Antiquity - what Gibbon called the gradual fall - is really a story of Rome learning to deal with and interact with foreign groups in ways other than conquest.
I think that America, today, is suffering from a very similar form of hubris. There's a pervasive belief that no external force will ever be able to make a serious impact on the United States. I'm not talking, here, about a 9/11-style impact or a military defeat; I'm talking about a belief that our empire can operate unilaterally without triggering long-term repercussions, that our power is such that we can simply give orders to other countries and not attempt to maintain the more sophisticated relations that one needs with other nations of comparable power.
At this precise moment, it may be true that we have enough political power in the world to be able to act this unilaterally; but this sort of power is notoriously ephemeral, and when it fails we may find ourselves in a position of not having any mechanisms prepared to be able to deal with other countries. Those other countries won't be suffering from the same lack - they've been dealing with one another for quite some time. This might leave us in a frighteningly weak position.
So these are the basic senses in which I believe that there are parallels between the United States and Rome. They're not absolute parallels, as I hope I've made clear - but there's enough in common here that I think we need to carefully study their history, and work hard to avoid making the same mistakes that they did.
