The aforementioned article mention that those were supposed to be the first elections with a proportional system, that it went by in a context of tension because of Nazi Germany, and that the two parties running got eight and seven seats respectively, with no vote.
The sources given aren't on the web. So, what happened?
Since the institution of voting the parliament consisted of members of the Fortschrittliche Bürgerpartei (shortend to FPB, translated Progressive Citizen's Party) and the Christlich-Soziale Volkspartei (shortend to VP, translated Christian-Social People's Party) the latter of which had lost the 1930 election completely so all 15 seats were taken by members of the FPB. This started a discussion about a change to a proportional voting system, but in 1932 the VP could regain 2 seats). In 1936 the VP merged with the Liechtensteiner Heimatdienst (Lichtensteinian Homeland Service), a minor party, to form the Vaterländische Union (shortened to VU, translated Patriotic Union) and gained another 2 seats.
By 1938 Austria had joined the Reich (the so called Anschluss) and suddenly Lichtenstein was a direct neighbor to the country which was aggressively putting the German Question on the table again. In Lichtenstein the Volksdeutsche Bewegung in Liechtenstein (German People's Movement in Lichtenstein) was founded and later attempted two coups in march of 1939. First by the burning of swastika flags hoping Hitler would invade, then by declaring an Anschluss, hoping to echo the events in Austria. The membership of 150 to 250 doesn't sound very high but if you compare it with the about 3000 eligible voters at the time it makes sense that the established parties feared the newcomer.
The talk about a proportional voting system had never died down and now led to the two parties in parliament trying to implement it to cool down it's proponents without letting the more radical parties into the parliament and to appease the Swiss neighbors (there was a customs union with the Swiss that had been questioned by the Anschluss proponents). So they compromised, formed a coalition for the first time and implemented a clause barring any party with less than 18% of the vote from sending any members. Additionally the new voting law (passed 1939-01-11) allowed the so called silent vote in article 30, if the parties comprising 80% of the last parliamentary election formed a common list of candidates and not enough eligible voters (at least 400) raised an objection to this list in a span of 14 days. This rule was then used to present a list of candidates on 1939-03-17. Then the aforementioned attempted coup took place and the coalition started a collection of signatures for the following text:
"Die Unterzeichneten erklären hiermit mit eigenhändiger Unterschrift feierlich, dass sie geschlossen auf dem Boden der Selbständigkeit und Unabhängigkeit unseres Heimatlandes unter dem Fürstenhaus von Liechtenstein und der Beibehaltung der Wirtschaftsverträge mit der Schweiz stehen und jedwede politische und wirtschaftliche Neuorientierung strikte ablehnen."
"The undersigned declare […] that they are standing on the foundation of self-reliance and independence of our home country under the Dynasty of Liechtenstein […] and oppose any new direction in respect to politics or economy."
They collected the signatures of 95,4% of eligible voters.
The new parliament was consolidated on 1939-04-13 and the coalition continued to rule together until 1997. The silent vote rule was never used again and was later removed as the voting law was replaced.
Available online sources (all in German):