Who had a higher standard of living, peasants in the Roman Empire, or peasants during the High Medieval era?

This is a sound question although the argument that cheap grain from Egypt undercut the peasents is deeply suspect.  The problem was always land ownership and security.  

Grain was shipped to key population centers and were consumed right there.  The surrounds could never have supplied all that grain at all.  Land transport costs were also a lot higher as well.

What killed the peasant was a steady imperial supply of slaves for the larger landholders who replacedthe peasent.  The owners  were not growing cash crops to ship either which made Southern Slavery bearable to their compatriots.

Thus the latifundia worked at self sufficincy with slaves and selling product to the cities in order to collect roman coinage.  No idea of fiat currency either, let alone banking.

The steady elimination of slavery in Europe led to growing peasantry and middle class.  It was a transition, but one in which all had some level of access to.  ...

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Who had a higher standard of living, peasants in the Roman Empire, or peasants during the High Medieval era?

Without a shred of doubt peasants in the High Medieval Period. 

No slavery, meaning the peasants are not undercut by an extremely cheap workforce.

Additionally, the peasant “class” in Ancient Rome was almost non-existant, there existed large landowners who owned most of the land and let slaves do their work, it was very difficult for a peasant to exist in those circumstances. 

No cheap grain from North Africa ( Tunis, Egypt ), highly fertile lands which compete with the farmers of Europe who lose profit. 

The Romans were terrible innovators… Already in the early Medieval Period did Peasants transition to a Two-Field crop rotation which made their land not only sustainable, but also more productive.

Yet in the High Medieval Period, we are speaking of Three-field crop rotation, much more productivity and also severely increasing the nutritional input of the population. 

Speaking of nutrition, the average medieval European was about 2cm taller than the average Roman which is a good indicator at how well fed the people were.

This would also mean a medieval peasant had a better life than a roman citizen. 

Medieval Peasants had the heavy-plough which allowed forests and resistent soil to be farmed. Allowing peasants to cultivate more land. 

Romans still used oxen, whereas in the early medieval period the ox was replaced with the horse as primary farm-animal. This was possible with inventions such as the horsecollar and the widespread adoption of the horseshoe. Horses travel much faster and are easier to control and with those inventions the lifespan of the animal is increased.

All of this increased productivity and also makes the lifes of peasants more comfortable and cheaper. 

Water mills were already known to the Romans, but they were not widespread, widespread adoption happened during the medieval times and not only that those high medieval europeans invented ( vertical ) wind-mills, harnessing the industrial power of water and wind to process grain.

Those types of peasants, the millers, were usually the richest peasants, there didn`t even exist an equivalent of a miller in the Roman Empire ( afterall why bother, when you can just use cheap slave labor? ). 

Now a comparison of pictures, first the average house of a Roman peasant : 

Nothing more than a small shack… 

Now here the high medieval peasant house : 

This striking difference should tell you everything you need to know. ( edit : though you should probably pretend the chimney doesn’t exist. ) 

Roman peasants were a small, declining demographic, whereas medieval peasants were flourishing.