Identify the greatest criticism of those who claim membership to either of the two major parties have of minor parties.
The largest criticism of third parties is they often skew election results. Had Ralph Nader not been involved in the 2000 election; it is entirely possible that Al Gore would have been President. Those who voted for Nader would have voted for Gore had they no choice but Gore and Bush. By creating a third alternative which is not a truly viable alternative, election results often are skewed such that the losing candidate--quite often a loser in the popular vote--will win the electoral vote (if the election is for President) and thereby the election.
The biggest thing that I hear people talk about against the third party is that they are not going to be able to actually win an election, so, as stated above, the votes are essentially wasted. I think that we may be on the verge of seeing this argument disappear as the Tea Party gains strength and popularity.
If the examination of the major poltical parties' responses against the minor ones is reflective of the United States political scene, the primary critique might be that the minor parties simply do not have enough political sway to make a legitimate difference. From a national political point of view, this is reflective of the "throw away" type of vote. If someone voted for the Green party, as an example, their vote would be one of protest. While the individual might actually believe the ideas of the party, the mathematical and political likelihood that the Green party would win resides in percentage points. Major political parties like the Democrats and Republicans have done a fairly good job of crowding out the field so that minor parties are true to the adjective. In the end, major parties argue that if one believes in workable solutions that can be accomplished, voting for the smaller parties is not politically feasible in a two system configuration like that of the United States.
The answer to this question depends to some extent on who you are asking. Different people have different criticisms and there is no "official" greatest criticism. Perhaps the most important criticism of third parties is that they encourage people to think about single issues instead of the "bigger picture."
When people vote for or claim membership in small parties, they are generally doing so because of a narrow range of issues. For example, people who join the Green Party are likely motivated mainly by environmental issues. This means that they are only caring about a single issue area insted of basing their vote on a broad range of ideas (everything from abortion to foreign policy, for example).
People who care mostly about party politics also criticize small party members for taking votes away from the big parties. A Democrat might say, for example, that Green Party voters in Florida in the 2000 Presidential election ruined the chances of the Democratic candidate. The Green voters threw away their votes on a candidate who had no chance. By doing so, they helped to elect Pres. Bush. People who care mainly about having their (big) party win would criticize the small parties for that as well.