Well...sadly the moment we speak there is only ONE party in the Parliament in favour (at least theoretically) of the monarchy, the National Liberal Party. They form an alliance with a small party, the Conservative Party. The two of them are allied with the third, the Social-Democrate Party, forming a strong political alliance that now managed to pull down the former government through a motion of confidence in the Parliament and they are now forming the new government.
There is also the Democrate-Liberal Party which has been governing since the end of 2008, now very impopular and most likely to lose severely at the next elections. There would be also the party of the hungarian minority, a smaller one, and...how should I say it... I'm really ashame of that, but there is a "party" now in the Parliament consisting of deputies and senators who "ran" from their original parties, under misterious presures (penal files most likely - most of the prosecutors are politically controlable), from the coalition in power (they were then in the opposition) and from the actual president of the country (he needed a comfortable majority in the Parliament to be on "his side", so that he wouldn't be suspended again like in 2007).
In november there will be parlamentary elections, most probably the coalition formed by the three parties I've mentioned at the beginning will obtain a score between 55 and 65 % of the votes, maybe even more, now with the new election law (just like the election law in Great Britain). The rest of the parties that are supposed to enter the new Parliament cannot be suspected of monarchist simpathies. Only some of their members in particular maybe, like T. Paleologu, who was supposed to read the petition in the Deputies Chamber.
The Social-Democrate Party is officially not a monarchist one, although it can be assumed that it has also monarchist members. The responsibility of a future restoration will be an issue of the Liberal Party. The president of the party, Crin Antonescu is a declared monarchist, but he seeks to become president of the republic. No one knows what he will do after he manages to get elected as president (he is by far the great favourite for that)...if he would support the restoration or not.
Basicly, if the social-democrates could be convinced by their partners the liberals of the necessity of restoring the monarchy...things would become extremely simple. They will be forming a large parliamentary majority, could be even 70%...and then they could easily write a new constitution, with the form of government changed.