All forms of legislative government have problems: the party political prone to them just as much as any other system.
The main quandaries with parties are that in grouping together to vote on legislation (as well as other executive facilities in certain jurisdictions) democracy itself falls by the wayside, due to key decisions being voted down the party line and not with the minds of the people who voted them in. However, because of this perceived lack of democracy, it's easier to pass crucial legislation if more people are going to blindly follow/reject it… sorta like a dictatorship!
As well as this, different parties have different levels of internal discipline. In the UK, for example, Parliamentarians pretty much have to keep to the party line or face expulsion from the party; in the US Congressional system, there's still more of a connection between the electorate and their Representatives/Senators despite all the corruption you hear of, so party discipline is reduced; however, it is still there on some of the most important issues (although the most polarizing issues can just cause havoc with the notion of parties at all, as has been mentioned in previous posts!)
In places where the parties in the legislative essentially elect the executive – as in Parliamentary systems – the added advantage of getting rid of them altogether would mean that the electorate may feel like they are actually having a say in both executive and legislative elections, rather than voting for a party for executive power and just ending up with a member of that same party for their legislative representative. However, especially in Europe, the fragmented nature of their party system meant that it can often take an awfully long time to even form a government after an election: the worst example is Belgium after their last election in 2010 where, 290 days later, a government STILL hasn't been formed due to parties clashing with each other.
However, the chief advantage of the party system, especially in modern politics – although it's possible to argue that this is caused by rather than an advantage of party politics – is that you have access to the party machine both throughout your election campaign as well as your time in office. Publicity tours, think tanks and even smear campaigns are all at your disposal as a party member, and you never know when you might have to use them! On top of this, the costs for getting elected in the first place have skyrocketed in recent years: and will continue to do so in the US after a recent Supreme Court ruling. (Again… yes this does sound kinda dictatorial.)
Finally, in areas where party discipline is relatively strong, and a two-party system is in de facto effect, the electorate as a whole can becomes pigeonholed into specific boxes on certain issues. The notion that one can't vote for anyone else can rise up, and force people to vote for the 'lesser of two evils' instead of a candidate that may better reflect their political positions… it's a wonder many people feel as cynical about politics as they do, not only in this thread, but in the wider world.
OK… that was longer than I had anticipated. To answer the question of whether parties should be abolished: it would be a nice idea and would undoubtedly lead to a more representative legislative procedure, but probably unworkable in modern day politics, mainly due to the funding mechanisms inherently provided, but also because people have become conditioned to the nature of party politics and may not like what they don't know. Better the devil you know than the devil you don't may apply here…
