According to the 3D Test of Antisemitism as defined by Natan Sharansky and adopted by the U.S. State Department, deligitimisation of Israel is antisemitic:
The term "delegitimization of Israel" refers to the denial of the Jewish people's right to self-determination, for example, by claiming that the existence of a State of Israel is a racist endeavor.
From this definition, it follows that Anti-Zionism is antisemitic. Many on the (far) left side in the Labour Party might describe themselves as Anti-Zionist.
The International Holocaust Remembrance Alliance considers that comparing Israeli policy with Nazi policy is antisemitic:
Drawing comparisons of contemporary Israeli policy to that of the Nazis.
It is not hard to find leftist individuals who make comparisons between (for example) the Israeli occupation of Palestine and the Nazi occupation of European countries, in particular in the heat of a debate.
Many leftist Jewish organisations disagree with the 3D Test or the IHRA example, and have defined their own rival definitions or examples. A proposed Labour definition does not consider comparing Israeli policy with Nazi policy as antisemitic, for which Labour has been criticised. Examples of lefist Jewish organisations who may be described as antizionist include Jewish Voice for Peace, Jews for Justice for Palestinians, or Jewdas with its alternative proposed definition, where Jewdas (in their own words) list “Jewish organisations who oppose the Zionist ideology and Israeli occupation and apartheid”. Jewish Voice for Labour in its proposed definition considers that “comparing Israel’s actions to that of the Nazis should not automatically be seen as anti-Semitic”. In this article in Die Welt (in German), a survey on attitudes in members of German political parties distinguishes between classical antisemitism and antisemitism in relation to Israel.
The Campaign against antisemitism has documented 39 cases of what they consider to be antisemitism within the Labour Party. It makes no distinction between “classical” antisemitism and ”antisemitism in relation to Israel”. Most of those incidents involve local councillors. Some involve what many would consider “classical” antisemitic stereotypes such as related to worldwide Jewish conspiracies. Others involve statements that are rather ”antisemitism in relation to Israel”: antisemitic according to the 3D Test or the IHRA example, but not according to the rival definitions such as from Jewdas or Jewish Voice for Labour.
For an example of a somewhat prominent incident that is antisemitic according to the 3D Test but not according to rival definitions, consider the incident where Naz Shah shared an image suggesting that if Israel was located in the middle of the USA, the Palestinian-Israeli conflict would be solved. Essentially, the gist of the message is that Israel as a Jewish state should not have been founded in the Middle East. This is antisemitic by the 3D Test definition, because it delegitimises Israel, even though many Jews are anti-Zionist as well.
P.S. The problem is prominent in the Labour Party, but not completely absent from other parties. Based on stats from the Campaign against Antisemitism, Labour is the 3rd worst prominent party considering incidents normalised by membership or number of councillors (most incidents involve councillors). Those stats are based on the mainstream definitions and determined by the Campaign against Antisemitism.
I don't know how representative this is.
- Labour Party: 39 incidents, 552,000 members, 6,470 councillors = 7.1 incidents/100k members or 6.0 incidents/1k councillors
- Conservative Party: 4 incidents, 124,000 members, 9,116 councillors = 3.2 incidents/100k members or 0.4 incidents/1k councillors
- Liberal-Democrats: 5 incidents, 100,500 members, 1,890 councillors = 5.0 incidents/100k members or 2.6 incidents/1k councillors
- Green Party of England and Wales: 5 incidents, 41,000 members, 180 councillors = 12.2 incidents/100k members or 27.8 incidents/1k councillors
- Scottish National Party: 2 incidents, 118,000 members, 423 councillors: 1.7 incidents/100k members or 4.7 incidents/1k councillors
- UKIP: 5 incidents, 21,000 members, 108 councillors, 23.8 incidents/100k members or 46 incidents/1k councillors councillors.
- Not documented: Sinn Féin, Plaid Cymru, Democratic Unionist Party