How much truth is in that?

None. I don't know who claims that? Unless you mean since the founding of the current Icelandic state in 1944. Iceland sometimes claims to have the world's oldest continually-operating parliament, but even that is a pretty big stretch.

Iceland was settled during the Viking Age, beginning in the late 9th century. The country was not ruled by a single king - but neither were any of the other would-be states in Scandinavia. Those - to the extent they were governed at all- were governed by local assemblies (þing or 'thing' - same word in singular/plural), where justice was enacted and decisions were made by vote. Although references to 'kings' go far back, there no written sources from Viking Age Scandinavia beyond short inscriptions. It's very unclear how many petty kings and kingdoms existed, and to what extent they held political and religious power. Most likely, political power was not ordained by law, or dynasty or god (or the gods) but a straightforward combination of wealth, of kinship and allies and other influence, influence from personal traits, charisma, likability, the ability to inspire loyalty, and influence that was simply bought - a person of high social standing was expected to display generosity, hold banquets, give gifts, and his status was contingent on his ability to do so. This is something that comes across fairly consistently in the post-Viking Age sagas that take place in it.

Towards the end of the Viking Age (11th century), unified countries - Norway, Denmark and Sweden begin to emerge. The kings and other magnates have not only converted their countries - either violently as in Norway or more gradually and peacefully as in Sweden. Iceland, we are told, was officially converted with a þing decision there in AD 1000.

So several things are going on as Scandinavia enters its Middle Ages (1050-1500) - kings have consolidated the country but they still wield fairly limited power, and that power is in no small part connected to a large number of royal estates. The kings were elected - and could be deposed - by the þing. The þing made laws, and kings vowed to uphold the laws upon election by the þing. And these would often be provincial þing with province-level laws. (e.g. Scania, Jutland, West-Götaland, Hordaland etc) Iceland being a small place had a single set of laws during the Commonwealth period, known as the Gray Goose (Grágás).

Although kingship was often dynastic within a small number of powerful families, hereditary kingship would not be formally instituted until the Reformation in the 1500s, which also saw a much greater centralization of power. These are events that defined the end of the Middle Ages, and effectively ended conflict that existed throughout the period.

To go back to the beginning; following the conversion to Christianity came the creation of Christian institutions in the continental mold (even if Scandinavia never really instituted feudalism, serfdom or strict political hierarchies). A new pole of power had been created - the Catholic Church. Bishops wielded great power; and perhaps even more so in Iceland, which had two dioceses (Skálholt and Hólar) by the 1100s, which was a lot on a per-capita basis. The late 11th century bishop of Hólar, Jón Ögmundsson, was powerful and influential enough to rename the week-days from their Pagan names and have it stick. This (and Jón) was puritanical even by medieval-bishop standards (Jón also disapproved of music and dancing). Most held to the words of Saint Isidore in his Etymologies, where he stated that although it would be preferable to have number days (except for Sabbath - Saturday, and the Lord's day, Sunday) analogous to Hebrew, that Pagan names were still tolerable. That wasn't good enough for Jón though, and Icelandic is the only Germanic language where the days were renamed. (And Portuguese the only Romance language, IIRC)

Nobility did in fact not exist in Viking Age or early Medieval Scandinavia. This was a concept that was introduced and formalized around the mid-late 1200s. But if the what of the nobility was a novel concept, then the _who- of it certainly wasn't : Those who joined those ranks were essentially the same wealth and well-connected people that'd been magnates earlier.

(Now somebody's going to say "Wait, but I read in a book on Norse mythology an origin story about karls (free men), jarls (nobles) and thralls (slaves)." - Yes this story is from Rígsþula, the oldest copy of which is from the early 1300s. And probably the strongest - but far from the only - argument for what that story is not from the Viking Age despite being about Heimdallr and such, is precisely this anachronism - it's more reflective of 13th century social structure than 10th. Even the term jarl itself is not attested from the actual Viking Age)

The 12th and first half of the 13th centuries saw turmoil and warfare in Scandinavia between powerful families and their allies; those siding with the king would of course claim to be acting out of fealty, while those siding with the opposition would claim the the king was overstepping his authority to their detriment. The 'Civil War Era' in Norway stretches from the 1130s to 1240s. In Sweden the Folkungs went into armed rebellion against the king five times in the 13th century alone, as well as strifed with the competing House of Sverker. Results differed, in Sweden and Denmark the nobility remained strong relative the king, while in Norway they were subjugated. A fact that would have relevance centuries later.

Iceland, since its founding until the mid 13th century had thus been a commonwealth, with no king. But that did not keep it out of the conflict; King Hákon of Norway was seeking to expand his influence over country, and so-called Sturlung Age of the early 13th century saw decades of strife between the Sturlungs and other clans, with varying loyalty and opposition to the Norwegians; this - and the Icelandic commonwealth - ended with the "Old Covenant" of 1264, which brought Iceland under the king of Norway. And as said, in Norway the king was powerful.

By 1400, Norway, Sweden and Denmark would have joined in a personal union, the Kalmar Union. However, already by the early 1434, Swedes had revolted and deposed king Erik of Pomerania. Only to later bring him back after concessions, then depose him again, bring him back again, and so on a number of times. Sweden's nobility strongly opposed the continuous efforts from the Denmark-based union kings to expand their own and Danish power over Sweden, ultimately leaving the union for good in 1521. The weakened status of the Norwegian nobility and Council of the Realm would (to simplify) lead to Norway (and thus Iceland) remaining in the union, already in 1450 the Bergen Treaty had been signed, declaring the two countries would have the same monarch 'in perpetuity'.

This evolved into direct Danish rule over Norway and Iceland, and the rule itself was absolute for a long time before Denmark-Norway formally became an absolute monarchy in 1665. Only in the 19th century would Denmark have a representative assembly passing laws again. Norway's Storting was created in 1814, while Sweden's Riksdag of the Estates (being separate chambers of representatives of the Nobility, Clergy, Burghers and Peasantry) had actually had continual legislative power since it was created in that form in 1435. (assemblies prior to that had become restricted to the clergy and nobility alone)

So the argument for the continuity of the Icelandic Alþingi rests on the fact that it was not abolished at that time, or any other. However, there would not have been any reason to abolish it at that time, as it had long since ceased functioning as a legislative parliament. Through the Old Covenant in the 1200s, legislative power was jointly held by the king and þing. But as said, the Norwegian kings held most of this power. This would only increase on behalf of the Danish kings, until the 1662 Kópavogur Treaty, where Icelanders formally relinquished all autonomy.