The source for this information is Medieval Warfare by Hans Delbruck.
According to Annals of Parma Frederick II had 10,000 men
Frederick set up his fortified camp opposite the smaller part of the city of Parma on the left bank of the Torrente Parma. The camp was famously called Vittoria. From here Frederick’s forces could lay waste to the area around Parma and seek to prevent reinforcements. Frederick’s likely strategy would have been to starve Parma into submission.
The city not surrounded. To cut off the city completely would have required forts of circumference of five miles with approx 40,000 men needed to guard them. Even if forts had been placed at points around the city it is likely that a force larger than the one Frederick had available would have been required. The English chronicler Matthew Paris states, however, that there were a number of castles around the city, bearing the name Vittoria, so maybe Frederick did engage in this strategy.
Unfortunately I have been unable to find any historical or archaeological sources that describe the possible location of Frederick’s camp. Looking at modern maps of Parma it would appear that the terrain around Parma apart from the river that cuts through it was fairly level undulating hills, so it is unlikely that any particular geographical features played a large party in the position of his camp or the battle.
It seems that the siege was not being particularly successful as Parma had significant forces available. For instance the Mantuans brought assistance via the river Po to the north. However, Matthew Paris in his chronicle does related that the Parmese were suffering and that they even asked Frederick for peace, an offer that he rejected, warning them that they must use there corn sparingly as he would not give them any more while he lived.
In the winter of 1247-1248 Frederick released the contingents from Bergamo, Pavia, Tortona and Allesandria and detached some of his troops to Treviso and Allesandria.
The forces Remaining were 1,100 horse and 2000 foot from Cremona, and Saracen archers. With a total of maybe 5000.
When the Parmese sallied out on 18 February 1248 Frederick was hunting with probably around 500 horsemen. The Parmese intended to move up the Po against Frederick’s son Enzo. Half of their forces did so and the other half became embroiled in an unintended battle with the Imperial forces. Again Matthew Paris is at odds with this version of events, he claims that the Parmese knew Frederick was absent, and that they planned the attack at this time to move against his camp. One expects that this could be true as surely they would be able to see Frederick leave with his large hunting party.
The Imperial forces lost and the Parmese entered the camp of Vittoria. Parmese claimed 1500 Imperials killed and 3000 captured. The Placentine annals say only 100 knights and 1500 foot soldiers were killed or captured however. One of the dead was Frederick’s minister Thaddeus de Suessa. Frederick also lost his treasury and other items of his baggage.
Could Frederick have succeeded in starving Parma if he had not suffered the ill-luck of being absent when the Parmese sallied? The Parmese did not know of his absence and Frederick’s forces engaged without any real orders apparently. If Frederick was there it is quite probable that defeat would not have occurred and that the siege would have been maintained.