What a good question! I believe it has a surprisingly simple answer: it reflects Hungary's personal union with the Kingdom of Poland (via its Jagiellon ruler).

As you're well aware, the Piast eagle is also known as the "Polish eagle"; it has been the regnal symbol of the Polish crown since 1295 and remains the emblem of the Polish Republic to this day. By hanging the Polish eagle inside the Országház, Hungary advertises its union not just with a Jagiellon, but with the crown of Poland.

Question for you, OP: did you assume (with good reason) that the banner was the Piast eagle, or did a tour guide/plaque/other educational material expressly tell you it was the Piast eagle?

• If you assumed that it was the Piast eagle, then I believe my answer is correct, it actually represents Poland as a sovereign nation, not merely the Piast and/or Jagiellon families.

• If, however, someone expressly told you that the banner was the Piast eagle, then I have more research to do. Why would a Jagiellon want to be represented by a Piast symbol, or why would the modern Hungarian state think a Jagiellon would want a Piast symbol? (Which is, of course, OP's original question.)

Fun with heraldry

During the first Jagiellon dynasty in Hungary (1440-1444), the royal coat of arms looked like this:

Władysław III Jagiellion coat of arms

The top left quarter includes the Árpád red/white stripes, the top right has the Piast/Polish eagle, the bottom left has the Jagiellon knight with the gold double cross on his blue shield, and the bottom right has the Magyar white double cross on red from King Béla III Árpád (r. 1172-1196).

During the second Jagiellon era in Hungary (1490-1526), the Hungarian royal coat of arms looked like this:

Louis II Jagiellon coat of arms

The left half has the Árpád red/white stripes, and the right half has the Magyar white double cross on red. I find it really interesting that during the second Jagiellon dynasty, the kings abandoned symbols of their Polish heritage in favor of fully Hungarian symbols.

The point of showing you the Hungarian Jagiellon symbols is that even during their rule, these kings did not center the Jagiellon gold double cross on blue. They actually didn't emphasize the Polish eagle either, but gave equal space or preference to traditional symbols of Hungarian royalty.

A little bit of history

In 1438, Albert von Habsburg was king of Hungary by marriage to Elizabeth of Luxembourg. When Albert died in 1439, his queen Elizabeth was pregnant. A civil war ensued: Elizabeth declared herself queen regent on behalf of her unborn son with the help of a powerful faction of Hungarian nobles. At the same time, other Hungarian nobles offered the Hungarian crown to Władysław III Jagiellon, King of Poland.

The reason some Hungarian nobles offered Władysław III the Hungarian crown was because they feared the ever-increasing threat of the Ottoman Empire's incursions into the Balkans. By 1440, the Ottomans had captured all of Greece, Bulgaria, and Serbia. Within 13 years, they would take Constantinople. (In 1440, Władysław III was only 16 years old. The Hungarians offered him the crown not for his personal strength/leadership, but to gain the might of Poland against the Ottomans.)

Under these circumstances, it's not surprising that Hungary wants to remember the Jagiellon dynasty not for the family itself, but because Poland came to Hungary's rescue when the Ottomans menaced the Magyars' southern border.

I'll repeat again what I said at the top: it is my educated guess that the banner you saw in the Országház was not the Piast eagle, but the symbol of the Polish crown. This speculation makes sense to me because of (a) the Jagiellon heraldry from that dynasty and (b) the geopolitical events that brought about a Polish king of Hungary in 1440.