From what it looks like on the California Secretary of State page, if you declare "No Party Preference (NPP)," you can vote in a party's primary if the party allows it. The Democratic Party has allowed it.
However if you belong to another party, you can't vote in the primary. The Independent Party seems to be signing up voters who have no party preference and may be intending to vote in the Democratic Party primary.
However, NPP voters may vote in a political party's partisan election if the political party, by party rule duly noticed to the Secretary of State, authorizes NPP voters to vote in the next presidential primary election. An NPP voter may request the ballot of one of the political parties, if any, that authorizes NPP voters to vote in the presidential primary election.
The following parties have notified the Secretary of State that they will allow No Party Preference voters to request their party’s presidential ballot in the March 3, 2020, Presidential Primary Election:
American Independent Party