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If the father is not learned in the Vedas, even then the son is allowed to study them.
Manu Smriti 3.136. If there is a father ignorant of the sacred texts whose son has learned one whole recension of the Veda and the Angas, and a son ignorant of the sacred texts whose father knows an entire recension of the Veda and the Angas,
Manu Smriti 3.137. Know that he whose father knows the Veda, is the more venerable one (of the two); yet the other one is worthy of honour, because respect is due to the Veda (which he has learned).
In 3.137, the other one refers to the son who is learned in the Vedas but whose father is not. Now, even such a son is worthy of respect.
So, the answer to your question is no.
This Shloka has nothing to do with studying of Vedas by father or son, it's wrongly interpreted above, read this:
योऽनधीत्य द्विजो वेदमन्यत्र कुरुते श्रमम् ।
स जीवन्नेव शूद्रत्वमाशु गच्छति सान्वयः ॥ मनुस्मृति । २ । १६८ ॥
मनु स्मृति के दूसरे अध्याय के १६८ वें श्लोक के अनुसार जिस द्विज ब्राह्मण ने वेद व वेदांग शास्त्र के अध्ययन के बदले धर्म निरपेक्ष व धर्म हीन शिक्षा ग्रहण की है वह ब्राह्मण से च्युत होकर अपने ही जीवन काल में , अपने वंश सहित, शूद्र वर्ण को प्राप्त करता है।
According to Manu Smriti: The twice born Brahmin who instead of studying Vedas and Ved Shastras, studies profane or faithless material, then that Brahmin (along with his descendents) in the present life cycle falls to the state of the śūdra.
read more here I think we should not always go by the way things have been translated in english.
Some what similar meaning here:
३) योऽनधीत्य द्विजो वेदमन्यत्र कुरुते श्रमम् | स जीवन्नेव शुद्रत्वमाशु गच्छति सान्वय: || (मनुस्मृति २/१६८)
अर्थ-जो द्विज अर्थात् ब्राह्मण, क्षत्रिय, वैश्य वेदाध्ययन का त्याग कर अन्य विद्याओं में ही परिश्रम करता रहता है, वह जीते-जी अपने आश्रित परिजनों के सहित शूद्रता को प्राप्त हो जाता है। क्योंकि, उसका ब्राह्मणत्व प्रदान करने वाला वेदाध्ययन छूटने से उसके आश्रित परिजनों का भी छूट जाता है, अतः वह परिवार शूद्र कहलाता है।
Means: The twice born Brahmin, Kshatriya, Vaishaya who gives up studying Vedas and continues to labour in other fields, then that person (along with his descendents) in the present life cycle falls to the state of the śūdra. hence such a family is referred as śūdra parivaar (family)
