Contrary to your belief, the current Chinese government (People's Republic of China, AKA PRC) isn't the first Chinese regime claimed the Spartly islands and Paracel islands. It all goes back to the 9-dash-line, which was originally the 11-dash-line...and it was published by Republic of China AKA ROC (previous regime, but still exist on their final stronghold on Taiwan, yes Taiwan's official name is still Republic of China) on the official map of China before the communists took over in 1949. Both Spartly islands and Paracel islands are within the 9-dash-line. At the time no other country claimed Spartly nor Paracel, nor anyone objected to China's claim except for the French who used to occupy Indochina (which Vietnam was a part of) as colonies.
Vietnam later claimed these islands based on the claim of French.
UNCLOS didn't even exist back then. EEZ didn't exist back then. If these two rules apply to all, then Guam wouldn't be belong to the US, and Falklands wouldn't be UK's.
Philippines claim also came in no earlier than 1960s.
You have to take note that there are 4 countries (Vietnam, Philiphines, Brunei, Malaysisa) other than China (Both PRC on mainland and ROC in Taiwan) claim Spartly islands. So who is right anyway? There can only be one "rightful" owner... right?
As for Paracel islands, if you look at the map, they are right off China's Hainan Province. Yes, that giant island province with 9 million population. Vietnam is further away from it.
The trick is, the US media always use propaganda against China. So all you hear is China "bullying". Every time when a dispute is mentioned in the media, China is always the "bad guy", and all the other countries are "defenseless".
What about Philippines navy shot and killed a Taiwanese fishman near Spartly? Then they turned and ran when a Taiwanese warship showed up. Was there any word of "bullying" used on Philiphines?
Not to mention Diaoyu/Senkuka islands dispute. Japan has stronger navy and airforce than China. And they are in actual control of the disputed islands despite the fact the islands are much closer to China than it is to Japan. Worse, Japan was portrayed as a bullied victim too by the biased media while China is clearly the weaker side between the two.
My advice to you.
Do not fall for the propaganda. Most western propaganda are very anti-China. You won't hear anything positive about China.